| L(s) = 1 | − 56·3-s − 7.86e3·5-s + 9.10e4·7-s + 9.45e4·9-s − 1.59e5·11-s − 1.05e6·13-s + 4.40e5·15-s + 1.43e6·17-s + 2.18e7·19-s − 5.09e6·21-s + 3.58e7·23-s + 1.30e7·25-s − 2.03e7·27-s + 2.28e8·29-s + 6.47e7·31-s + 8.90e6·33-s − 7.16e8·35-s − 7.55e7·37-s + 5.88e7·39-s + 1.20e9·41-s + 4.55e7·43-s − 7.43e8·45-s − 1.22e9·47-s + 3.05e9·49-s − 8.01e7·51-s + 3.80e9·53-s + 1.25e9·55-s + ⋯ |
| L(s) = 1 | − 0.133·3-s − 1.12·5-s + 2.04·7-s + 0.533·9-s − 0.297·11-s − 0.784·13-s + 0.149·15-s + 0.244·17-s + 2.02·19-s − 0.272·21-s + 1.16·23-s + 0.267·25-s − 0.272·27-s + 2.07·29-s + 0.406·31-s + 0.0396·33-s − 2.30·35-s − 0.179·37-s + 0.104·39-s + 1.61·41-s + 0.0472·43-s − 0.600·45-s − 0.781·47-s + 1.54·49-s − 0.0325·51-s + 1.25·53-s + 0.335·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4096 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4096 ^{s/2} \, \Gamma_{\C}(s+11/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(4.416137773\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.416137773\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 + 56 T - 10154 p^{2} T^{2} + 56 p^{11} T^{3} + p^{22} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 7868 T + 9768358 p T^{2} + 7868 p^{11} T^{3} + p^{22} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 13008 p T + 106936526 p^{2} T^{2} - 13008 p^{12} T^{3} + p^{22} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 159080 T + 362536063286 T^{2} + 159080 p^{11} T^{3} + p^{22} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 1050476 T + 3458956762062 T^{2} + 1050476 p^{11} T^{3} + p^{22} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 1430884 T + 16748384809766 T^{2} - 1430884 p^{11} T^{3} + p^{22} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 21866600 T + 343123710088422 T^{2} - 21866600 p^{11} T^{3} + p^{22} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 35806736 T + 1952928132896462 T^{2} - 35806736 p^{11} T^{3} + p^{22} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 228827700 T + 35907977791027054 T^{2} - 228827700 p^{11} T^{3} + p^{22} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 64722112 T + 51498184379920062 T^{2} - 64722112 p^{11} T^{3} + p^{22} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 75558780 T + 354292341022528382 T^{2} + 75558780 p^{11} T^{3} + p^{22} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 1201214196 T + 1274442257818453270 T^{2} - 1201214196 p^{11} T^{3} + p^{22} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 45519832 T + 42144714696540642 p T^{2} - 45519832 p^{11} T^{3} + p^{22} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 1229079264 T + 5024390553242310430 T^{2} + 1229079264 p^{11} T^{3} + p^{22} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 3808549924 T + 16648234587904299038 T^{2} - 3808549924 p^{11} T^{3} + p^{22} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 6012926584 T + 61066799326040273366 T^{2} - 6012926584 p^{11} T^{3} + p^{22} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 9789792908 T + \)\(10\!\cdots\!42\)\( T^{2} + 9789792908 p^{11} T^{3} + p^{22} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 14703095224 T + 95414710392374126214 T^{2} + 14703095224 p^{11} T^{3} + p^{22} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 4319991088 T + \)\(28\!\cdots\!82\)\( T^{2} - 4319991088 p^{11} T^{3} + p^{22} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 11055639476 T + \)\(65\!\cdots\!62\)\( T^{2} - 11055639476 p^{11} T^{3} + p^{22} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 51957623264 T + \)\(14\!\cdots\!78\)\( T^{2} - 51957623264 p^{11} T^{3} + p^{22} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 108227975912 T + \)\(54\!\cdots\!06\)\( T^{2} - 108227975912 p^{11} T^{3} + p^{22} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 71188291860 T + \)\(31\!\cdots\!82\)\( T^{2} - 71188291860 p^{11} T^{3} + p^{22} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 1699807676 T + \)\(12\!\cdots\!50\)\( T^{2} + 1699807676 p^{11} T^{3} + p^{22} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.64791168998141460591703571494, −12.13941003232605515200950729752, −11.58395380307863217202890752055, −11.52247107968015188924707920459, −10.66906195660690869240479172452, −10.23977169119348969378462158462, −9.393801435770576858822691500217, −8.742567131935207824343908151892, −7.939187171239743310401124711566, −7.66195628930883171209764100030, −7.37180581703682958816235720022, −6.40263699948488029341276526281, −5.24717884400475488011806296267, −4.88925930324107622293152501822, −4.52554364297864246683120781738, −3.56663466846543407004933338445, −2.79729362559968037129079242923, −1.90835091319347185472830877432, −0.900666369998735598325841053243, −0.810502374803978014274553916560,
0.810502374803978014274553916560, 0.900666369998735598325841053243, 1.90835091319347185472830877432, 2.79729362559968037129079242923, 3.56663466846543407004933338445, 4.52554364297864246683120781738, 4.88925930324107622293152501822, 5.24717884400475488011806296267, 6.40263699948488029341276526281, 7.37180581703682958816235720022, 7.66195628930883171209764100030, 7.939187171239743310401124711566, 8.742567131935207824343908151892, 9.393801435770576858822691500217, 10.23977169119348969378462158462, 10.66906195660690869240479172452, 11.52247107968015188924707920459, 11.58395380307863217202890752055, 12.13941003232605515200950729752, 12.64791168998141460591703571494
