| L(s) = 1 | − 537.·5-s − 343·7-s + 2.41e3·11-s + 5.19e3·13-s + 2.57e4·17-s − 3.41e4·19-s + 9.35e4·23-s + 2.10e5·25-s + 1.70e5·29-s − 2.85e5·31-s + 1.84e5·35-s − 5.40e5·37-s − 2.19e5·41-s + 8.76e5·43-s − 5.32e5·47-s + 1.17e5·49-s − 1.70e6·53-s − 1.29e6·55-s − 2.00e5·59-s − 9.41e5·61-s − 2.78e6·65-s − 2.43e6·67-s + 4.15e6·71-s + 5.31e6·73-s − 8.29e5·77-s − 3.41e6·79-s + 5.76e6·83-s + ⋯ |
| L(s) = 1 | − 1.92·5-s − 0.377·7-s + 0.547·11-s + 0.655·13-s + 1.27·17-s − 1.14·19-s + 1.60·23-s + 2.69·25-s + 1.29·29-s − 1.72·31-s + 0.726·35-s − 1.75·37-s − 0.497·41-s + 1.68·43-s − 0.747·47-s + 0.142·49-s − 1.57·53-s − 1.05·55-s − 0.126·59-s − 0.531·61-s − 1.25·65-s − 0.989·67-s + 1.37·71-s + 1.59·73-s − 0.207·77-s − 0.778·79-s + 1.10·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + 343T \) |
| good | 5 | \( 1 + 537.T + 7.81e4T^{2} \) |
| 11 | \( 1 - 2.41e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 5.19e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 2.57e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 3.41e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 9.35e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.70e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.85e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 5.40e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 2.19e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 8.76e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 5.32e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.70e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 2.00e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 9.41e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.43e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 4.15e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 5.31e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 3.41e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 5.76e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 5.37e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 7.26e6T + 8.07e13T^{2} \) |
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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
−10.64691483109122279997255709783, −9.104695967605969306161676816503, −8.344970414571502587369838789719, −7.38897798401130006308341552390, −6.53580384491488422215690756300, −4.97208592264084290486570141198, −3.82574341543921192698948856484, −3.19372785401240981995850190577, −1.13582202255186179223433697523, 0,
1.13582202255186179223433697523, 3.19372785401240981995850190577, 3.82574341543921192698948856484, 4.97208592264084290486570141198, 6.53580384491488422215690756300, 7.38897798401130006308341552390, 8.344970414571502587369838789719, 9.104695967605969306161676816503, 10.64691483109122279997255709783
