| L(s) = 1 | + 5-s − 3.03·7-s − 5.27·11-s − 0.513·13-s + 2.80·17-s − 8.29·19-s − 5.03·23-s + 25-s − 4.78·29-s + 8.58·31-s − 3.03·35-s − 6.58·37-s + 7.98·41-s + 1.19·43-s + 9.62·47-s + 2.22·49-s − 0.467·53-s − 5.27·55-s + 0.757·59-s − 0.271·61-s − 0.513·65-s + 7.26·67-s − 1.48·71-s + 5.31·73-s + 16.0·77-s + 15.9·79-s − 12.0·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.14·7-s − 1.58·11-s − 0.142·13-s + 0.679·17-s − 1.90·19-s − 1.05·23-s + 0.200·25-s − 0.888·29-s + 1.54·31-s − 0.513·35-s − 1.08·37-s + 1.24·41-s + 0.182·43-s + 1.40·47-s + 0.317·49-s − 0.0642·53-s − 0.710·55-s + 0.0985·59-s − 0.0347·61-s − 0.0637·65-s + 0.887·67-s − 0.176·71-s + 0.622·73-s + 1.82·77-s + 1.79·79-s − 1.32·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.025670107\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.025670107\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| good | 7 | \( 1 + 3.03T + 7T^{2} \) |
| 11 | \( 1 + 5.27T + 11T^{2} \) |
| 13 | \( 1 + 0.513T + 13T^{2} \) |
| 17 | \( 1 - 2.80T + 17T^{2} \) |
| 19 | \( 1 + 8.29T + 19T^{2} \) |
| 23 | \( 1 + 5.03T + 23T^{2} \) |
| 29 | \( 1 + 4.78T + 29T^{2} \) |
| 31 | \( 1 - 8.58T + 31T^{2} \) |
| 37 | \( 1 + 6.58T + 37T^{2} \) |
| 41 | \( 1 - 7.98T + 41T^{2} \) |
| 43 | \( 1 - 1.19T + 43T^{2} \) |
| 47 | \( 1 - 9.62T + 47T^{2} \) |
| 53 | \( 1 + 0.467T + 53T^{2} \) |
| 59 | \( 1 - 0.757T + 59T^{2} \) |
| 61 | \( 1 + 0.271T + 61T^{2} \) |
| 67 | \( 1 - 7.26T + 67T^{2} \) |
| 71 | \( 1 + 1.48T + 71T^{2} \) |
| 73 | \( 1 - 5.31T + 73T^{2} \) |
| 79 | \( 1 - 15.9T + 79T^{2} \) |
| 83 | \( 1 + 12.0T + 83T^{2} \) |
| 89 | \( 1 - 15.2T + 89T^{2} \) |
| 97 | \( 1 + 6.36T + 97T^{2} \) |
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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
−8.021482132583163011864089826935, −7.33700277114591428513542666760, −6.44742526129524158683036536015, −5.97814247553088364390317751231, −5.29728476669164893721153059566, −4.38460558315350216167745181507, −3.55197066434517151432097471000, −2.61771070816209598691420058924, −2.11166606127574900519902338700, −0.48845638341821851160548601542,
0.48845638341821851160548601542, 2.11166606127574900519902338700, 2.61771070816209598691420058924, 3.55197066434517151432097471000, 4.38460558315350216167745181507, 5.29728476669164893721153059566, 5.97814247553088364390317751231, 6.44742526129524158683036536015, 7.33700277114591428513542666760, 8.021482132583163011864089826935
