| L(s) = 1 | − 1.83·2-s + 3-s + 1.36·4-s + 5-s − 1.83·6-s + 0.959·7-s + 1.16·8-s + 9-s − 1.83·10-s − 0.930·11-s + 1.36·12-s + 6.87·13-s − 1.76·14-s + 15-s − 4.86·16-s + 7.27·17-s − 1.83·18-s − 6.06·19-s + 1.36·20-s + 0.959·21-s + 1.70·22-s − 2.05·23-s + 1.16·24-s + 25-s − 12.6·26-s + 27-s + 1.30·28-s + ⋯ |
| L(s) = 1 | − 1.29·2-s + 0.577·3-s + 0.682·4-s + 0.447·5-s − 0.748·6-s + 0.362·7-s + 0.412·8-s + 0.333·9-s − 0.580·10-s − 0.280·11-s + 0.393·12-s + 1.90·13-s − 0.470·14-s + 0.258·15-s − 1.21·16-s + 1.76·17-s − 0.432·18-s − 1.39·19-s + 0.305·20-s + 0.209·21-s + 0.364·22-s − 0.427·23-s + 0.237·24-s + 0.200·25-s − 2.47·26-s + 0.192·27-s + 0.247·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.710951277\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.710951277\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 - T \) |
| good | 2 | \( 1 + 1.83T + 2T^{2} \) |
| 7 | \( 1 - 0.959T + 7T^{2} \) |
| 11 | \( 1 + 0.930T + 11T^{2} \) |
| 13 | \( 1 - 6.87T + 13T^{2} \) |
| 17 | \( 1 - 7.27T + 17T^{2} \) |
| 19 | \( 1 + 6.06T + 19T^{2} \) |
| 23 | \( 1 + 2.05T + 23T^{2} \) |
| 29 | \( 1 - 8.89T + 29T^{2} \) |
| 31 | \( 1 - 3.39T + 31T^{2} \) |
| 37 | \( 1 - 3.06T + 37T^{2} \) |
| 41 | \( 1 + 6.29T + 41T^{2} \) |
| 43 | \( 1 - 5.27T + 43T^{2} \) |
| 47 | \( 1 + 3.83T + 47T^{2} \) |
| 53 | \( 1 - 6.27T + 53T^{2} \) |
| 59 | \( 1 + 3.14T + 59T^{2} \) |
| 61 | \( 1 + 3.48T + 61T^{2} \) |
| 67 | \( 1 - 7.11T + 67T^{2} \) |
| 71 | \( 1 - 1.18T + 71T^{2} \) |
| 73 | \( 1 - 16.9T + 73T^{2} \) |
| 79 | \( 1 - 14.0T + 79T^{2} \) |
| 83 | \( 1 + 9.25T + 83T^{2} \) |
| 89 | \( 1 + 5.69T + 89T^{2} \) |
| 97 | \( 1 - 17.8T + 97T^{2} \) |
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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
−8.107865992870499359952069109953, −7.968701547948084627259178644529, −6.71670753690766185436556004595, −6.27203547430691839265249249205, −5.27187936415123373810573235994, −4.33756817944192201361840779876, −3.51812860449037038973913341884, −2.49839116481240657655332365198, −1.54187283977858925128011386571, −0.912087222873914476381965358376,
0.912087222873914476381965358376, 1.54187283977858925128011386571, 2.49839116481240657655332365198, 3.51812860449037038973913341884, 4.33756817944192201361840779876, 5.27187936415123373810573235994, 6.27203547430691839265249249205, 6.71670753690766185436556004595, 7.968701547948084627259178644529, 8.107865992870499359952069109953
