| L(s) = 1 | − 5·5-s − 16·7-s + 36·11-s − 42·13-s + 110·17-s + 116·19-s + 16·23-s + 25·25-s − 198·29-s − 240·31-s + 80·35-s − 258·37-s − 442·41-s + 292·43-s + 392·47-s − 87·49-s − 142·53-s − 180·55-s − 348·59-s − 570·61-s + 210·65-s − 692·67-s + 168·71-s − 134·73-s − 576·77-s − 784·79-s + 564·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.863·7-s + 0.986·11-s − 0.896·13-s + 1.56·17-s + 1.40·19-s + 0.145·23-s + 1/5·25-s − 1.26·29-s − 1.39·31-s + 0.386·35-s − 1.14·37-s − 1.68·41-s + 1.03·43-s + 1.21·47-s − 0.253·49-s − 0.368·53-s − 0.441·55-s − 0.767·59-s − 1.19·61-s + 0.400·65-s − 1.26·67-s + 0.280·71-s − 0.214·73-s − 0.852·77-s − 1.11·79-s + 0.745·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + p T \) |
| good | 7 | \( 1 + 16 T + p^{3} T^{2} \) |
| 11 | \( 1 - 36 T + p^{3} T^{2} \) |
| 13 | \( 1 + 42 T + p^{3} T^{2} \) |
| 17 | \( 1 - 110 T + p^{3} T^{2} \) |
| 19 | \( 1 - 116 T + p^{3} T^{2} \) |
| 23 | \( 1 - 16 T + p^{3} T^{2} \) |
| 29 | \( 1 + 198 T + p^{3} T^{2} \) |
| 31 | \( 1 + 240 T + p^{3} T^{2} \) |
| 37 | \( 1 + 258 T + p^{3} T^{2} \) |
| 41 | \( 1 + 442 T + p^{3} T^{2} \) |
| 43 | \( 1 - 292 T + p^{3} T^{2} \) |
| 47 | \( 1 - 392 T + p^{3} T^{2} \) |
| 53 | \( 1 + 142 T + p^{3} T^{2} \) |
| 59 | \( 1 + 348 T + p^{3} T^{2} \) |
| 61 | \( 1 + 570 T + p^{3} T^{2} \) |
| 67 | \( 1 + 692 T + p^{3} T^{2} \) |
| 71 | \( 1 - 168 T + p^{3} T^{2} \) |
| 73 | \( 1 + 134 T + p^{3} T^{2} \) |
| 79 | \( 1 + 784 T + p^{3} T^{2} \) |
| 83 | \( 1 - 564 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1034 T + p^{3} T^{2} \) |
| 97 | \( 1 + 382 T + p^{3} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.543690975066072429623702159999, −8.951891471947032040563394274677, −7.52451760798479621278663508109, −7.22608319743094168985827171129, −5.96538394294430222702470974443, −5.11847996101791448743214064276, −3.73582908767111393171100873805, −3.13875424788702571891199556941, −1.44263295184550130017139136020, 0,
1.44263295184550130017139136020, 3.13875424788702571891199556941, 3.73582908767111393171100873805, 5.11847996101791448743214064276, 5.96538394294430222702470974443, 7.22608319743094168985827171129, 7.52451760798479621278663508109, 8.951891471947032040563394274677, 9.543690975066072429623702159999
