L(s) = 1 | + 2-s − 3.13·3-s + 4-s − 0.529·5-s − 3.13·6-s + 7-s + 8-s + 6.82·9-s − 0.529·10-s − 3.29·11-s − 3.13·12-s − 0.516·13-s + 14-s + 1.65·15-s + 16-s + 6.14·17-s + 6.82·18-s − 0.529·20-s − 3.13·21-s − 3.29·22-s − 5.69·23-s − 3.13·24-s − 4.71·25-s − 0.516·26-s − 11.9·27-s + 28-s − 0.202·29-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.80·3-s + 0.5·4-s − 0.236·5-s − 1.27·6-s + 0.377·7-s + 0.353·8-s + 2.27·9-s − 0.167·10-s − 0.994·11-s − 0.904·12-s − 0.143·13-s + 0.267·14-s + 0.428·15-s + 0.250·16-s + 1.48·17-s + 1.60·18-s − 0.118·20-s − 0.683·21-s − 0.703·22-s − 1.18·23-s − 0.639·24-s − 0.943·25-s − 0.101·26-s − 2.30·27-s + 0.188·28-s − 0.0376·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + 3.13T + 3T^{2} \) |
| 5 | \( 1 + 0.529T + 5T^{2} \) |
| 11 | \( 1 + 3.29T + 11T^{2} \) |
| 13 | \( 1 + 0.516T + 13T^{2} \) |
| 17 | \( 1 - 6.14T + 17T^{2} \) |
| 23 | \( 1 + 5.69T + 23T^{2} \) |
| 29 | \( 1 + 0.202T + 29T^{2} \) |
| 31 | \( 1 + 0.736T + 31T^{2} \) |
| 37 | \( 1 + 10.8T + 37T^{2} \) |
| 41 | \( 1 - 3.68T + 41T^{2} \) |
| 43 | \( 1 - 7.19T + 43T^{2} \) |
| 47 | \( 1 - 12.1T + 47T^{2} \) |
| 53 | \( 1 - 3.18T + 53T^{2} \) |
| 59 | \( 1 + 3.91T + 59T^{2} \) |
| 61 | \( 1 - 3.73T + 61T^{2} \) |
| 67 | \( 1 + 9.77T + 67T^{2} \) |
| 71 | \( 1 - 6.75T + 71T^{2} \) |
| 73 | \( 1 - 16.3T + 73T^{2} \) |
| 79 | \( 1 + 8.69T + 79T^{2} \) |
| 83 | \( 1 + 18.0T + 83T^{2} \) |
| 89 | \( 1 + 16.2T + 89T^{2} \) |
| 97 | \( 1 - 2.63T + 97T^{2} \) |
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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
−7.50064693053033675749986318185, −7.15573274868689532201368524972, −5.99312226800186983486505700555, −5.67576495847649149475680848360, −5.16303973136434904052135192097, −4.32675250017400540121131152129, −3.66193891907975225550483679348, −2.32635647857266430353682969920, −1.22093321294781659505626938253, 0,
1.22093321294781659505626938253, 2.32635647857266430353682969920, 3.66193891907975225550483679348, 4.32675250017400540121131152129, 5.16303973136434904052135192097, 5.67576495847649149475680848360, 5.99312226800186983486505700555, 7.15573274868689532201368524972, 7.50064693053033675749986318185