L(s) = 1 | + 1.46·2-s − 3-s + 0.155·4-s + 2.96·5-s − 1.46·6-s + 7-s − 2.70·8-s + 9-s + 4.36·10-s + 0.0918·11-s − 0.155·12-s + 2.83·13-s + 1.46·14-s − 2.96·15-s − 4.28·16-s − 5.52·17-s + 1.46·18-s + 1.96·19-s + 0.463·20-s − 21-s + 0.134·22-s − 4.21·23-s + 2.70·24-s + 3.81·25-s + 4.15·26-s − 27-s + 0.155·28-s + ⋯ |
L(s) = 1 | + 1.03·2-s − 0.577·3-s + 0.0779·4-s + 1.32·5-s − 0.599·6-s + 0.377·7-s − 0.957·8-s + 0.333·9-s + 1.37·10-s + 0.0276·11-s − 0.0450·12-s + 0.785·13-s + 0.392·14-s − 0.766·15-s − 1.07·16-s − 1.34·17-s + 0.346·18-s + 0.449·19-s + 0.103·20-s − 0.218·21-s + 0.0287·22-s − 0.878·23-s + 0.552·24-s + 0.763·25-s + 0.815·26-s − 0.192·27-s + 0.0294·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 383 | \( 1 - T \) |
good | 2 | \( 1 - 1.46T + 2T^{2} \) |
| 5 | \( 1 - 2.96T + 5T^{2} \) |
| 11 | \( 1 - 0.0918T + 11T^{2} \) |
| 13 | \( 1 - 2.83T + 13T^{2} \) |
| 17 | \( 1 + 5.52T + 17T^{2} \) |
| 19 | \( 1 - 1.96T + 19T^{2} \) |
| 23 | \( 1 + 4.21T + 23T^{2} \) |
| 29 | \( 1 + 4.30T + 29T^{2} \) |
| 31 | \( 1 + 5.16T + 31T^{2} \) |
| 37 | \( 1 + 1.56T + 37T^{2} \) |
| 41 | \( 1 + 8.45T + 41T^{2} \) |
| 43 | \( 1 + 3.19T + 43T^{2} \) |
| 47 | \( 1 + 5.81T + 47T^{2} \) |
| 53 | \( 1 + 3.81T + 53T^{2} \) |
| 59 | \( 1 - 8.97T + 59T^{2} \) |
| 61 | \( 1 - 3.16T + 61T^{2} \) |
| 67 | \( 1 + 8.85T + 67T^{2} \) |
| 71 | \( 1 - 15.6T + 71T^{2} \) |
| 73 | \( 1 + 11.0T + 73T^{2} \) |
| 79 | \( 1 + 3.47T + 79T^{2} \) |
| 83 | \( 1 + 15.1T + 83T^{2} \) |
| 89 | \( 1 + 5.57T + 89T^{2} \) |
| 97 | \( 1 + 7.94T + 97T^{2} \) |
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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
−7.07277256064886225008020378269, −6.52435606017635381616793845125, −5.83020121091094813867456997855, −5.48920004210770103965631982024, −4.79067894400898259638749215640, −4.06851519438347836349944199936, −3.29750351494981999753558854756, −2.18423781729474393496590818803, −1.56049654333340475906287876182, 0,
1.56049654333340475906287876182, 2.18423781729474393496590818803, 3.29750351494981999753558854756, 4.06851519438347836349944199936, 4.79067894400898259638749215640, 5.48920004210770103965631982024, 5.83020121091094813867456997855, 6.52435606017635381616793845125, 7.07277256064886225008020378269