| L(s) = 1 | − 2.53·3-s + 5-s − 0.798·7-s + 3.44·9-s − 0.408·11-s − 3.81·13-s − 2.53·15-s − 3.49·17-s + 7.76·19-s + 2.02·21-s − 5.65·23-s + 25-s − 1.14·27-s − 5.07·29-s − 0.0918·31-s + 1.03·33-s − 0.798·35-s − 6.95·37-s + 9.70·39-s + 3.73·41-s − 0.431·43-s + 3.44·45-s + 8.30·47-s − 6.36·49-s + 8.88·51-s − 5.25·53-s − 0.408·55-s + ⋯ |
| L(s) = 1 | − 1.46·3-s + 0.447·5-s − 0.301·7-s + 1.14·9-s − 0.123·11-s − 1.05·13-s − 0.655·15-s − 0.848·17-s + 1.78·19-s + 0.442·21-s − 1.17·23-s + 0.200·25-s − 0.219·27-s − 0.943·29-s − 0.0164·31-s + 0.180·33-s − 0.134·35-s − 1.14·37-s + 1.55·39-s + 0.582·41-s − 0.0658·43-s + 0.514·45-s + 1.21·47-s − 0.908·49-s + 1.24·51-s − 0.722·53-s − 0.0551·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7012582368\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7012582368\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 151 | \( 1 + T \) |
| good | 3 | \( 1 + 2.53T + 3T^{2} \) |
| 7 | \( 1 + 0.798T + 7T^{2} \) |
| 11 | \( 1 + 0.408T + 11T^{2} \) |
| 13 | \( 1 + 3.81T + 13T^{2} \) |
| 17 | \( 1 + 3.49T + 17T^{2} \) |
| 19 | \( 1 - 7.76T + 19T^{2} \) |
| 23 | \( 1 + 5.65T + 23T^{2} \) |
| 29 | \( 1 + 5.07T + 29T^{2} \) |
| 31 | \( 1 + 0.0918T + 31T^{2} \) |
| 37 | \( 1 + 6.95T + 37T^{2} \) |
| 41 | \( 1 - 3.73T + 41T^{2} \) |
| 43 | \( 1 + 0.431T + 43T^{2} \) |
| 47 | \( 1 - 8.30T + 47T^{2} \) |
| 53 | \( 1 + 5.25T + 53T^{2} \) |
| 59 | \( 1 - 12.9T + 59T^{2} \) |
| 61 | \( 1 - 0.863T + 61T^{2} \) |
| 67 | \( 1 + 9.61T + 67T^{2} \) |
| 71 | \( 1 - 1.28T + 71T^{2} \) |
| 73 | \( 1 - 3.55T + 73T^{2} \) |
| 79 | \( 1 - 7.20T + 79T^{2} \) |
| 83 | \( 1 + 3.03T + 83T^{2} \) |
| 89 | \( 1 + 8.56T + 89T^{2} \) |
| 97 | \( 1 + 2.89T + 97T^{2} \) |
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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
−7.82313950001029432539014336151, −7.18604527883850163372725154317, −6.59201874673738318455686720150, −5.79836792757096850384401790058, −5.35485286209121828831017520002, −4.73239866896331880436976640802, −3.79125976677495670562556380037, −2.68899653894308647468781264985, −1.69643548936309666884237838333, −0.46897086900568735206927426812,
0.46897086900568735206927426812, 1.69643548936309666884237838333, 2.68899653894308647468781264985, 3.79125976677495670562556380037, 4.73239866896331880436976640802, 5.35485286209121828831017520002, 5.79836792757096850384401790058, 6.59201874673738318455686720150, 7.18604527883850163372725154317, 7.82313950001029432539014336151
