| L(s) = 1 | − 8.44·3-s + 5·5-s + 16.2·7-s + 44.3·9-s − 0.0653·11-s − 34.3·13-s − 42.2·15-s − 71.6·17-s + 34.2·19-s − 136.·21-s + 68.5·23-s + 25·25-s − 146.·27-s − 29·29-s − 280.·31-s + 0.552·33-s + 81.0·35-s + 403.·37-s + 289.·39-s − 318.·41-s − 273.·43-s + 221.·45-s + 518.·47-s − 80.1·49-s + 605.·51-s − 492.·53-s − 0.326·55-s + ⋯ |
| L(s) = 1 | − 1.62·3-s + 0.447·5-s + 0.875·7-s + 1.64·9-s − 0.00179·11-s − 0.732·13-s − 0.726·15-s − 1.02·17-s + 0.413·19-s − 1.42·21-s + 0.621·23-s + 0.200·25-s − 1.04·27-s − 0.185·29-s − 1.62·31-s + 0.00291·33-s + 0.391·35-s + 1.79·37-s + 1.19·39-s − 1.21·41-s − 0.969·43-s + 0.734·45-s + 1.60·47-s − 0.233·49-s + 1.66·51-s − 1.27·53-s − 0.000801·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| 29 | \( 1 + 29T \) |
| good | 3 | \( 1 + 8.44T + 27T^{2} \) |
| 7 | \( 1 - 16.2T + 343T^{2} \) |
| 11 | \( 1 + 0.0653T + 1.33e3T^{2} \) |
| 13 | \( 1 + 34.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 71.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 34.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 68.5T + 1.21e4T^{2} \) |
| 31 | \( 1 + 280.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 403.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 318.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 273.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 518.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 492.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 861.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 125.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 1.03e3T + 3.00e5T^{2} \) |
| 71 | \( 1 + 91.9T + 3.57e5T^{2} \) |
| 73 | \( 1 - 936.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 768.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 666.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 736.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.34e3T + 9.12e5T^{2} \) |
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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
−9.213782719806049952217996588694, −8.057866623359042349468440214382, −7.06327354768894779708528907706, −6.47323258162652762101611186582, −5.30884823049643253771071556349, −5.11988914680641522474192705937, −4.04856419690327727213677573137, −2.31540376413599944374847371773, −1.20320400037759451762651098439, 0,
1.20320400037759451762651098439, 2.31540376413599944374847371773, 4.04856419690327727213677573137, 5.11988914680641522474192705937, 5.30884823049643253771071556349, 6.47323258162652762101611186582, 7.06327354768894779708528907706, 8.057866623359042349468440214382, 9.213782719806049952217996588694
