L(s) = 1 | − 2-s − 2.14·3-s + 4-s − 5-s + 2.14·6-s + 3.62·7-s − 8-s + 1.59·9-s + 10-s − 5.65·11-s − 2.14·12-s − 0.701·13-s − 3.62·14-s + 2.14·15-s + 16-s + 5.75·17-s − 1.59·18-s − 2.95·19-s − 20-s − 7.76·21-s + 5.65·22-s + 2.14·24-s + 25-s + 0.701·26-s + 3.01·27-s + 3.62·28-s + 0.108·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.23·3-s + 0.5·4-s − 0.447·5-s + 0.874·6-s + 1.36·7-s − 0.353·8-s + 0.531·9-s + 0.316·10-s − 1.70·11-s − 0.618·12-s − 0.194·13-s − 0.968·14-s + 0.553·15-s + 0.250·16-s + 1.39·17-s − 0.375·18-s − 0.677·19-s − 0.223·20-s − 1.69·21-s + 1.20·22-s + 0.437·24-s + 0.200·25-s + 0.137·26-s + 0.580·27-s + 0.684·28-s + 0.0200·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 \) |
good | 3 | \( 1 + 2.14T + 3T^{2} \) |
| 7 | \( 1 - 3.62T + 7T^{2} \) |
| 11 | \( 1 + 5.65T + 11T^{2} \) |
| 13 | \( 1 + 0.701T + 13T^{2} \) |
| 17 | \( 1 - 5.75T + 17T^{2} \) |
| 19 | \( 1 + 2.95T + 19T^{2} \) |
| 29 | \( 1 - 0.108T + 29T^{2} \) |
| 31 | \( 1 + 2.03T + 31T^{2} \) |
| 37 | \( 1 + 2.92T + 37T^{2} \) |
| 41 | \( 1 - 8.21T + 41T^{2} \) |
| 43 | \( 1 - 7.57T + 43T^{2} \) |
| 47 | \( 1 + 10.2T + 47T^{2} \) |
| 53 | \( 1 + 6.83T + 53T^{2} \) |
| 59 | \( 1 + 5.41T + 59T^{2} \) |
| 61 | \( 1 - 7.06T + 61T^{2} \) |
| 67 | \( 1 + 15.0T + 67T^{2} \) |
| 71 | \( 1 - 10.1T + 71T^{2} \) |
| 73 | \( 1 - 1.89T + 73T^{2} \) |
| 79 | \( 1 - 9.07T + 79T^{2} \) |
| 83 | \( 1 - 15.2T + 83T^{2} \) |
| 89 | \( 1 - 2.95T + 89T^{2} \) |
| 97 | \( 1 - 0.0397T + 97T^{2} \) |
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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
−7.943486436955566843305640024381, −7.36678675577389299568147984206, −6.37493488101529442026568839225, −5.53207660605296466430230196917, −5.14383500293286863944501830605, −4.42151871124490510043247546866, −3.12992959547042982636463054979, −2.12168149499530784560320884244, −1.02553295340857869141668872611, 0,
1.02553295340857869141668872611, 2.12168149499530784560320884244, 3.12992959547042982636463054979, 4.42151871124490510043247546866, 5.14383500293286863944501830605, 5.53207660605296466430230196917, 6.37493488101529442026568839225, 7.36678675577389299568147984206, 7.943486436955566843305640024381