| L(s) = 1 | + 6·3-s − 10·5-s + 4·7-s + 27·9-s + 40·11-s − 24·13-s − 60·15-s + 32·17-s + 116·19-s + 24·21-s − 12·23-s + 75·25-s + 108·27-s + 148·29-s + 76·31-s + 240·33-s − 40·35-s + 280·37-s − 144·39-s + 572·41-s + 520·43-s − 270·45-s − 340·47-s + 130·49-s + 192·51-s + 524·53-s − 400·55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s + 0.215·7-s + 9-s + 1.09·11-s − 0.512·13-s − 1.03·15-s + 0.456·17-s + 1.40·19-s + 0.249·21-s − 0.108·23-s + 3/5·25-s + 0.769·27-s + 0.947·29-s + 0.440·31-s + 1.26·33-s − 0.193·35-s + 1.24·37-s − 0.591·39-s + 2.17·41-s + 1.84·43-s − 0.894·45-s − 1.05·47-s + 0.379·49-s + 0.527·51-s + 1.35·53-s − 0.980·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230400 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.360706636\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.360706636\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 - 4 T - 114 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 20 T + p^{3} T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 + 24 T + 3734 T^{2} + 24 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 32 T + 2846 T^{2} - 32 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 116 T + 16278 T^{2} - 116 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 12 T + 23566 T^{2} + 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 148 T + 41390 T^{2} - 148 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 76 T - 4098 T^{2} - 76 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 280 T + 100806 T^{2} - 280 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 572 T + 216422 T^{2} - 572 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 520 T + 213750 T^{2} - 520 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 340 T + 100670 T^{2} + 340 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 524 T + 337454 T^{2} - 524 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 552 T + 435478 T^{2} + 552 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 628 T + 163422 T^{2} - 628 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 168 T + 286982 T^{2} - 168 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 1248 T + 1053742 T^{2} + 1248 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 908 T + 723654 T^{2} - 908 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 1052 T + 1223358 T^{2} + 1052 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 40 T + 938150 T^{2} - 40 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 828 T + 857734 T^{2} - 828 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 316 T - 671034 T^{2} + 316 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−10.89195999420503365865128832469, −10.13254441819733875365628237018, −9.848017801791531731264736708663, −9.436387704973736674137603408515, −8.843653158631719744454767642130, −8.750249254305893253716540901777, −7.84063531446713307099307528031, −7.82974908836104763408757624992, −7.26295291946146576499045064391, −6.97944477624608067420526082003, −6.07212395300377907053598849885, −5.81501618418828311382857964648, −4.66855025925024018782452245963, −4.61855209234005759330070485959, −3.85729975159760863567950790838, −3.47336047385439846872753620141, −2.75729727742228247318676678477, −2.32746941148447204563904868637, −1.13623247091394193554145993441, −0.841081131810377554802056908417,
0.841081131810377554802056908417, 1.13623247091394193554145993441, 2.32746941148447204563904868637, 2.75729727742228247318676678477, 3.47336047385439846872753620141, 3.85729975159760863567950790838, 4.61855209234005759330070485959, 4.66855025925024018782452245963, 5.81501618418828311382857964648, 6.07212395300377907053598849885, 6.97944477624608067420526082003, 7.26295291946146576499045064391, 7.82974908836104763408757624992, 7.84063531446713307099307528031, 8.750249254305893253716540901777, 8.843653158631719744454767642130, 9.436387704973736674137603408515, 9.848017801791531731264736708663, 10.13254441819733875365628237018, 10.89195999420503365865128832469
