L(s) = 1 | − 2.35·2-s + 3.54·4-s − 5-s − 0.193·7-s − 3.64·8-s + 2.35·10-s − 0.973·13-s + 0.455·14-s + 1.49·16-s − 2.67·17-s + 5.54·19-s − 3.54·20-s + 4.80·23-s + 25-s + 2.29·26-s − 0.686·28-s − 10.1·29-s − 2.50·31-s + 3.76·32-s + 6.30·34-s + 0.193·35-s + 5.71·37-s − 13.0·38-s + 3.64·40-s − 8.27·41-s − 5.11·43-s − 11.3·46-s + ⋯ |
L(s) = 1 | − 1.66·2-s + 1.77·4-s − 0.447·5-s − 0.0731·7-s − 1.29·8-s + 0.744·10-s − 0.270·13-s + 0.121·14-s + 0.374·16-s − 0.648·17-s + 1.27·19-s − 0.793·20-s + 1.00·23-s + 0.200·25-s + 0.449·26-s − 0.129·28-s − 1.87·29-s − 0.449·31-s + 0.666·32-s + 1.08·34-s + 0.0327·35-s + 0.939·37-s − 2.12·38-s + 0.577·40-s − 1.29·41-s − 0.779·43-s − 1.66·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + 2.35T + 2T^{2} \) |
| 7 | \( 1 + 0.193T + 7T^{2} \) |
| 13 | \( 1 + 0.973T + 13T^{2} \) |
| 17 | \( 1 + 2.67T + 17T^{2} \) |
| 19 | \( 1 - 5.54T + 19T^{2} \) |
| 23 | \( 1 - 4.80T + 23T^{2} \) |
| 29 | \( 1 + 10.1T + 29T^{2} \) |
| 31 | \( 1 + 2.50T + 31T^{2} \) |
| 37 | \( 1 - 5.71T + 37T^{2} \) |
| 41 | \( 1 + 8.27T + 41T^{2} \) |
| 43 | \( 1 + 5.11T + 43T^{2} \) |
| 47 | \( 1 - 10.8T + 47T^{2} \) |
| 53 | \( 1 - 9.28T + 53T^{2} \) |
| 59 | \( 1 - 11.2T + 59T^{2} \) |
| 61 | \( 1 + 1.20T + 61T^{2} \) |
| 67 | \( 1 - 3.25T + 67T^{2} \) |
| 71 | \( 1 + 5.99T + 71T^{2} \) |
| 73 | \( 1 + 3.30T + 73T^{2} \) |
| 79 | \( 1 + 10.1T + 79T^{2} \) |
| 83 | \( 1 - 8.31T + 83T^{2} \) |
| 89 | \( 1 + 7.34T + 89T^{2} \) |
| 97 | \( 1 + 15.8T + 97T^{2} \) |
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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
−7.87919818544666193042249997160, −7.16909278259511723472814026509, −6.97506986273227424518421954717, −5.81350681491618359630059673421, −5.01326575956747728228448775745, −3.93006753137946633303443168379, −2.99189619479345113227055213284, −2.05597332459992670796535693505, −1.06529704109526876903059981105, 0,
1.06529704109526876903059981105, 2.05597332459992670796535693505, 2.99189619479345113227055213284, 3.93006753137946633303443168379, 5.01326575956747728228448775745, 5.81350681491618359630059673421, 6.97506986273227424518421954717, 7.16909278259511723472814026509, 7.87919818544666193042249997160