| L(s) = 1 | − 4.78·2-s − 5.99·3-s + 14.8·4-s + 28.6·6-s − 11.9·7-s − 32.7·8-s + 8.91·9-s − 11·11-s − 89.0·12-s − 22.4·13-s + 57.1·14-s + 37.8·16-s + 131.·17-s − 42.5·18-s + 99.0·19-s + 71.6·21-s + 52.5·22-s + 0.206·23-s + 196.·24-s + 107.·26-s + 108.·27-s − 177.·28-s − 163.·29-s + 217.·31-s + 81.3·32-s + 65.9·33-s − 630.·34-s + ⋯ |
| L(s) = 1 | − 1.69·2-s − 1.15·3-s + 1.85·4-s + 1.94·6-s − 0.645·7-s − 1.44·8-s + 0.330·9-s − 0.301·11-s − 2.14·12-s − 0.478·13-s + 1.09·14-s + 0.591·16-s + 1.88·17-s − 0.557·18-s + 1.19·19-s + 0.745·21-s + 0.509·22-s + 0.00187·23-s + 1.67·24-s + 0.808·26-s + 0.772·27-s − 1.19·28-s − 1.04·29-s + 1.25·31-s + 0.449·32-s + 0.347·33-s − 3.18·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 + 4.78T + 8T^{2} \) |
| 3 | \( 1 + 5.99T + 27T^{2} \) |
| 7 | \( 1 + 11.9T + 343T^{2} \) |
| 13 | \( 1 + 22.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 131.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 99.0T + 6.85e3T^{2} \) |
| 23 | \( 1 - 0.206T + 1.21e4T^{2} \) |
| 29 | \( 1 + 163.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 217.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 17.8T + 5.06e4T^{2} \) |
| 41 | \( 1 + 32.3T + 6.89e4T^{2} \) |
| 43 | \( 1 + 490.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 518.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 110.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 242.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 713.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 571.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 113.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 767.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 470.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.15e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 719.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 510.T + 9.12e5T^{2} \) |
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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
−10.66223381062421851381717165748, −10.02873684117439981694233221342, −9.338317822406418731122198895795, −8.019774517873546920009177244123, −7.24917997419879368221081199437, −6.18161772006151311220534969411, −5.21150253514238299909884249974, −3.03899168649630070547656221715, −1.16079970326181627380577536905, 0,
1.16079970326181627380577536905, 3.03899168649630070547656221715, 5.21150253514238299909884249974, 6.18161772006151311220534969411, 7.24917997419879368221081199437, 8.019774517873546920009177244123, 9.338317822406418731122198895795, 10.02873684117439981694233221342, 10.66223381062421851381717165748
