L(s) = 1 | − 2-s − 1.29·3-s + 4-s − 3.65·5-s + 1.29·6-s − 4.10·7-s − 8-s − 1.33·9-s + 3.65·10-s − 11-s − 1.29·12-s + 0.136·13-s + 4.10·14-s + 4.71·15-s + 16-s − 0.366·17-s + 1.33·18-s − 3.85·19-s − 3.65·20-s + 5.30·21-s + 22-s + 2.98·23-s + 1.29·24-s + 8.33·25-s − 0.136·26-s + 5.59·27-s − 4.10·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.745·3-s + 0.5·4-s − 1.63·5-s + 0.526·6-s − 1.55·7-s − 0.353·8-s − 0.444·9-s + 1.15·10-s − 0.301·11-s − 0.372·12-s + 0.0379·13-s + 1.09·14-s + 1.21·15-s + 0.250·16-s − 0.0888·17-s + 0.314·18-s − 0.883·19-s − 0.816·20-s + 1.15·21-s + 0.213·22-s + 0.621·23-s + 0.263·24-s + 1.66·25-s − 0.0268·26-s + 1.07·27-s − 0.776·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 197 | \( 1 + T \) |
good | 3 | \( 1 + 1.29T + 3T^{2} \) |
| 5 | \( 1 + 3.65T + 5T^{2} \) |
| 7 | \( 1 + 4.10T + 7T^{2} \) |
| 13 | \( 1 - 0.136T + 13T^{2} \) |
| 17 | \( 1 + 0.366T + 17T^{2} \) |
| 19 | \( 1 + 3.85T + 19T^{2} \) |
| 23 | \( 1 - 2.98T + 23T^{2} \) |
| 29 | \( 1 - 3.50T + 29T^{2} \) |
| 31 | \( 1 - 1.21T + 31T^{2} \) |
| 37 | \( 1 + 5.54T + 37T^{2} \) |
| 41 | \( 1 + 4.91T + 41T^{2} \) |
| 43 | \( 1 - 6.61T + 43T^{2} \) |
| 47 | \( 1 - 2.70T + 47T^{2} \) |
| 53 | \( 1 - 9.87T + 53T^{2} \) |
| 59 | \( 1 + 10.3T + 59T^{2} \) |
| 61 | \( 1 - 7.32T + 61T^{2} \) |
| 67 | \( 1 + 0.357T + 67T^{2} \) |
| 71 | \( 1 - 11.3T + 71T^{2} \) |
| 73 | \( 1 + 6.07T + 73T^{2} \) |
| 79 | \( 1 + 0.351T + 79T^{2} \) |
| 83 | \( 1 - 1.48T + 83T^{2} \) |
| 89 | \( 1 - 6.22T + 89T^{2} \) |
| 97 | \( 1 - 6.54T + 97T^{2} \) |
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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
−8.109639663864326435933836652943, −7.17045475970928291344872674797, −6.72678031649056952117209992317, −6.04663883605748049030864897783, −5.09926176323965325753211436011, −4.09260568116039499627948723243, −3.31822325832574917322698655070, −2.60443907242654286119073088445, −0.71633400037086649718130769333, 0,
0.71633400037086649718130769333, 2.60443907242654286119073088445, 3.31822325832574917322698655070, 4.09260568116039499627948723243, 5.09926176323965325753211436011, 6.04663883605748049030864897783, 6.72678031649056952117209992317, 7.17045475970928291344872674797, 8.109639663864326435933836652943