| L(s) = 1 | − 4.79·2-s + 3·3-s + 14.9·4-s − 14.3·6-s + 0.140·7-s − 33.4·8-s + 9·9-s + 47.3·11-s + 44.9·12-s + 34.3·13-s − 0.673·14-s + 40.4·16-s + 16.5·17-s − 43.1·18-s + 91.0·19-s + 0.421·21-s − 226.·22-s − 161.·23-s − 100.·24-s − 164.·26-s + 27·27-s + 2.10·28-s − 12.3·29-s − 333.·31-s + 73.7·32-s + 142.·33-s − 79.1·34-s + ⋯ |
| L(s) = 1 | − 1.69·2-s + 0.577·3-s + 1.87·4-s − 0.978·6-s + 0.00758·7-s − 1.47·8-s + 0.333·9-s + 1.29·11-s + 1.08·12-s + 0.732·13-s − 0.0128·14-s + 0.631·16-s + 0.235·17-s − 0.564·18-s + 1.09·19-s + 0.00438·21-s − 2.19·22-s − 1.46·23-s − 0.852·24-s − 1.24·26-s + 0.192·27-s + 0.0142·28-s − 0.0789·29-s − 1.92·31-s + 0.407·32-s + 0.749·33-s − 0.399·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - 3T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 4.79T + 8T^{2} \) |
| 7 | \( 1 - 0.140T + 343T^{2} \) |
| 11 | \( 1 - 47.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 34.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 16.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 91.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + 161.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 12.3T + 2.43e4T^{2} \) |
| 31 | \( 1 + 333.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 281.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 125.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 529.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 75.0T + 1.03e5T^{2} \) |
| 53 | \( 1 + 249.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 268.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 369.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 176.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 722.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 2.44T + 3.89e5T^{2} \) |
| 79 | \( 1 - 607.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.26e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.33e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.43e3T + 9.12e5T^{2} \) |
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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
−8.493492257549881420255431179775, −8.014694050729052205336272699362, −7.12445993023538940721572874895, −6.55430571203837018923794438665, −5.50476634130515248613241818155, −3.99681641702991681240320792674, −3.23276086001644648616086689213, −1.82301116344535834634666120481, −1.35265997760158039914150633527, 0,
1.35265997760158039914150633527, 1.82301116344535834634666120481, 3.23276086001644648616086689213, 3.99681641702991681240320792674, 5.50476634130515248613241818155, 6.55430571203837018923794438665, 7.12445993023538940721572874895, 8.014694050729052205336272699362, 8.493492257549881420255431179775
