| L(s) = 1 | + 2-s + 4-s + 3·5-s + 8-s + 3·10-s + 11-s − 6·13-s + 16-s − 5·17-s − 6·19-s + 3·20-s + 22-s − 5·23-s + 4·25-s − 6·26-s + 6·29-s − 4·31-s + 32-s − 5·34-s − 2·37-s − 6·38-s + 3·40-s + 5·41-s − 10·43-s + 44-s − 5·46-s + 9·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.34·5-s + 0.353·8-s + 0.948·10-s + 0.301·11-s − 1.66·13-s + 1/4·16-s − 1.21·17-s − 1.37·19-s + 0.670·20-s + 0.213·22-s − 1.04·23-s + 4/5·25-s − 1.17·26-s + 1.11·29-s − 0.718·31-s + 0.176·32-s − 0.857·34-s − 0.328·37-s − 0.973·38-s + 0.474·40-s + 0.780·41-s − 1.52·43-s + 0.150·44-s − 0.737·46-s + 1.31·47-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 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 9 T + p T^{2} \) |
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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
−7.05193844462886897025180878629, −6.51189120017659564869514313810, −6.01373658213606524536336613575, −5.25079894366371441975941867783, −4.60173741269309709378537458865, −4.04102461081452306682053960763, −2.78432011689669489335783274514, −2.25192768127746863667722000510, −1.68355529608785649414798664703, 0,
1.68355529608785649414798664703, 2.25192768127746863667722000510, 2.78432011689669489335783274514, 4.04102461081452306682053960763, 4.60173741269309709378537458865, 5.25079894366371441975941867783, 6.01373658213606524536336613575, 6.51189120017659564869514313810, 7.05193844462886897025180878629
