| L(s) = 1 | − 17·5-s + 35·7-s + 2·11-s + 13·13-s + 19·17-s − 94·19-s − 72·23-s + 164·25-s − 246·29-s + 100·31-s − 595·35-s − 11·37-s + 280·41-s − 241·43-s + 137·47-s + 882·49-s + 232·53-s − 34·55-s − 386·59-s + 64·61-s − 221·65-s + 670·67-s + 55·71-s − 838·73-s + 70·77-s − 1.01e3·79-s + 420·83-s + ⋯ |
| L(s) = 1 | − 1.52·5-s + 1.88·7-s + 0.0548·11-s + 0.277·13-s + 0.271·17-s − 1.13·19-s − 0.652·23-s + 1.31·25-s − 1.57·29-s + 0.579·31-s − 2.87·35-s − 0.0488·37-s + 1.06·41-s − 0.854·43-s + 0.425·47-s + 18/7·49-s + 0.601·53-s − 0.0833·55-s − 0.851·59-s + 0.134·61-s − 0.421·65-s + 1.22·67-s + 0.0919·71-s − 1.34·73-s + 0.103·77-s − 1.44·79-s + 0.555·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 - p T \) |
| good | 5 | \( 1 + 17 T + p^{3} T^{2} \) |
| 7 | \( 1 - 5 p T + p^{3} T^{2} \) |
| 11 | \( 1 - 2 T + p^{3} T^{2} \) |
| 17 | \( 1 - 19 T + p^{3} T^{2} \) |
| 19 | \( 1 + 94 T + p^{3} T^{2} \) |
| 23 | \( 1 + 72 T + p^{3} T^{2} \) |
| 29 | \( 1 + 246 T + p^{3} T^{2} \) |
| 31 | \( 1 - 100 T + p^{3} T^{2} \) |
| 37 | \( 1 + 11 T + p^{3} T^{2} \) |
| 41 | \( 1 - 280 T + p^{3} T^{2} \) |
| 43 | \( 1 + 241 T + p^{3} T^{2} \) |
| 47 | \( 1 - 137 T + p^{3} T^{2} \) |
| 53 | \( 1 - 232 T + p^{3} T^{2} \) |
| 59 | \( 1 + 386 T + p^{3} T^{2} \) |
| 61 | \( 1 - 64 T + p^{3} T^{2} \) |
| 67 | \( 1 - 10 p T + p^{3} T^{2} \) |
| 71 | \( 1 - 55 T + p^{3} T^{2} \) |
| 73 | \( 1 + 838 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1016 T + p^{3} T^{2} \) |
| 83 | \( 1 - 420 T + p^{3} T^{2} \) |
| 89 | \( 1 - 934 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1154 T + p^{3} T^{2} \) |
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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
−8.355522368061648951025045534180, −7.75910669480615604202353314592, −7.26841828752314731384713815953, −6.02109663669785878417350379345, −5.01771348357223877325515643800, −4.26919348557776827373593119672, −3.74395799217437517923055615446, −2.29275526584971034223335309243, −1.24766391959122719253345319765, 0,
1.24766391959122719253345319765, 2.29275526584971034223335309243, 3.74395799217437517923055615446, 4.26919348557776827373593119672, 5.01771348357223877325515643800, 6.02109663669785878417350379345, 7.26841828752314731384713815953, 7.75910669480615604202353314592, 8.355522368061648951025045534180
