| L(s) = 1 | − 1.52·2-s − 3·3-s − 5.66·4-s + 4.57·6-s − 4.84·7-s + 20.8·8-s + 9·9-s − 61.0·11-s + 17.0·12-s − 13·13-s + 7.39·14-s + 13.5·16-s + 41.7·17-s − 13.7·18-s − 107.·19-s + 14.5·21-s + 93.2·22-s − 28.5·23-s − 62.5·24-s + 19.8·26-s − 27·27-s + 27.4·28-s − 89.8·29-s + 183.·31-s − 187.·32-s + 183.·33-s − 63.7·34-s + ⋯ |
| L(s) = 1 | − 0.539·2-s − 0.577·3-s − 0.708·4-s + 0.311·6-s − 0.261·7-s + 0.922·8-s + 0.333·9-s − 1.67·11-s + 0.409·12-s − 0.277·13-s + 0.141·14-s + 0.211·16-s + 0.596·17-s − 0.179·18-s − 1.29·19-s + 0.150·21-s + 0.903·22-s − 0.258·23-s − 0.532·24-s + 0.149·26-s − 0.192·27-s + 0.185·28-s − 0.575·29-s + 1.06·31-s − 1.03·32-s + 0.966·33-s − 0.321·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.2785645045\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2785645045\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 + 13T \) |
| good | 2 | \( 1 + 1.52T + 8T^{2} \) |
| 7 | \( 1 + 4.84T + 343T^{2} \) |
| 11 | \( 1 + 61.0T + 1.33e3T^{2} \) |
| 17 | \( 1 - 41.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 107.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 28.5T + 1.21e4T^{2} \) |
| 29 | \( 1 + 89.8T + 2.43e4T^{2} \) |
| 31 | \( 1 - 183.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 418.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 142.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 71.0T + 7.95e4T^{2} \) |
| 47 | \( 1 + 323.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 25.1T + 1.48e5T^{2} \) |
| 59 | \( 1 + 684.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 308.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 672.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 326.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 24.3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 166.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 201.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 108.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.15e3T + 9.12e5T^{2} \) |
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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
−9.845914264877445996689232576221, −8.728908234258878937542791213083, −8.044088649337617145793913385223, −7.29123023624269806777806037155, −6.14105362303135450855300664961, −5.18388382881376232628976559980, −4.56131868485139247528104614236, −3.26726472797994589614640172109, −1.84100622850104537477689917515, −0.30971744606056264015873050238,
0.30971744606056264015873050238, 1.84100622850104537477689917515, 3.26726472797994589614640172109, 4.56131868485139247528104614236, 5.18388382881376232628976559980, 6.14105362303135450855300664961, 7.29123023624269806777806037155, 8.044088649337617145793913385223, 8.728908234258878937542791213083, 9.845914264877445996689232576221
