| L(s) = 1 | + 2·2-s + 2·3-s + 4·4-s + 4·6-s + 26·7-s + 8·8-s − 23·9-s − 28·11-s + 8·12-s + 12·13-s + 52·14-s + 16·16-s − 64·17-s − 46·18-s − 60·19-s + 52·21-s − 56·22-s − 58·23-s + 16·24-s + 24·26-s − 100·27-s + 104·28-s + 90·29-s − 128·31-s + 32·32-s − 56·33-s − 128·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.384·3-s + 1/2·4-s + 0.272·6-s + 1.40·7-s + 0.353·8-s − 0.851·9-s − 0.767·11-s + 0.192·12-s + 0.256·13-s + 0.992·14-s + 1/4·16-s − 0.913·17-s − 0.602·18-s − 0.724·19-s + 0.540·21-s − 0.542·22-s − 0.525·23-s + 0.136·24-s + 0.181·26-s − 0.712·27-s + 0.701·28-s + 0.576·29-s − 0.741·31-s + 0.176·32-s − 0.295·33-s − 0.645·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.223203766\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.223203766\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p^{3} T^{2} \) |
| 7 | \( 1 - 26 T + p^{3} T^{2} \) |
| 11 | \( 1 + 28 T + p^{3} T^{2} \) |
| 13 | \( 1 - 12 T + p^{3} T^{2} \) |
| 17 | \( 1 + 64 T + p^{3} T^{2} \) |
| 19 | \( 1 + 60 T + p^{3} T^{2} \) |
| 23 | \( 1 + 58 T + p^{3} T^{2} \) |
| 29 | \( 1 - 90 T + p^{3} T^{2} \) |
| 31 | \( 1 + 128 T + p^{3} T^{2} \) |
| 37 | \( 1 - 236 T + p^{3} T^{2} \) |
| 41 | \( 1 - 242 T + p^{3} T^{2} \) |
| 43 | \( 1 - 362 T + p^{3} T^{2} \) |
| 47 | \( 1 - 226 T + p^{3} T^{2} \) |
| 53 | \( 1 + 108 T + p^{3} T^{2} \) |
| 59 | \( 1 + 20 T + p^{3} T^{2} \) |
| 61 | \( 1 - 542 T + p^{3} T^{2} \) |
| 67 | \( 1 + 434 T + p^{3} T^{2} \) |
| 71 | \( 1 + 1128 T + p^{3} T^{2} \) |
| 73 | \( 1 - 632 T + p^{3} T^{2} \) |
| 79 | \( 1 + 720 T + p^{3} T^{2} \) |
| 83 | \( 1 + 478 T + p^{3} T^{2} \) |
| 89 | \( 1 + 490 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1456 T + p^{3} T^{2} \) |
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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
−14.77129951890050027381751937900, −14.09227179545508539474353914012, −12.95716080246762732420210290515, −11.52575913777725730280932238924, −10.74453963380820217789484345268, −8.726195892201281257052379545762, −7.69841516160078407168830209981, −5.81584695497405257785655748273, −4.41591693684014353406097660278, −2.37376225911289568237317825355,
2.37376225911289568237317825355, 4.41591693684014353406097660278, 5.81584695497405257785655748273, 7.69841516160078407168830209981, 8.726195892201281257052379545762, 10.74453963380820217789484345268, 11.52575913777725730280932238924, 12.95716080246762732420210290515, 14.09227179545508539474353914012, 14.77129951890050027381751937900
