| L(s) = 1 | − 2-s + 4-s − 8-s + 11-s − 2·13-s + 16-s − 6·17-s − 2·19-s − 22-s − 5·25-s + 2·26-s + 6·29-s + 4·31-s − 32-s + 6·34-s + 2·37-s + 2·38-s + 6·41-s − 10·43-s + 44-s + 12·47-s + 5·50-s − 2·52-s + 12·53-s − 6·58-s + 12·59-s + 10·61-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s + 0.301·11-s − 0.554·13-s + 1/4·16-s − 1.45·17-s − 0.458·19-s − 0.213·22-s − 25-s + 0.392·26-s + 1.11·29-s + 0.718·31-s − 0.176·32-s + 1.02·34-s + 0.328·37-s + 0.324·38-s + 0.937·41-s − 1.52·43-s + 0.150·44-s + 1.75·47-s + 0.707·50-s − 0.277·52-s + 1.64·53-s − 0.787·58-s + 1.56·59-s + 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−7.22256681200124913942068687753, −6.86319691613869780218188863622, −6.13488512587456810544110169825, −5.40474200258705758947525517831, −4.41604514672277597522072986408, −3.93582844976437782803839905400, −2.64565898313481655439998497825, −2.25338768348159295239588485846, −1.09696957382779221676747338783, 0,
1.09696957382779221676747338783, 2.25338768348159295239588485846, 2.64565898313481655439998497825, 3.93582844976437782803839905400, 4.41604514672277597522072986408, 5.40474200258705758947525517831, 6.13488512587456810544110169825, 6.86319691613869780218188863622, 7.22256681200124913942068687753
