L(s) = 1 | − 2-s + 4-s + 2.59·5-s + 0.164·7-s − 8-s − 2.59·10-s − 0.573·13-s − 0.164·14-s + 16-s − 2.30·19-s + 2.59·20-s + 0.409·23-s + 1.70·25-s + 0.573·26-s + 0.164·28-s − 8.48·29-s + 5.48·31-s − 32-s + 0.425·35-s − 7.19·37-s + 2.30·38-s − 2.59·40-s − 0.284·41-s − 6.28·43-s − 0.409·46-s − 9.96·47-s − 6.97·49-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.15·5-s + 0.0620·7-s − 0.353·8-s − 0.819·10-s − 0.159·13-s − 0.0438·14-s + 0.250·16-s − 0.529·19-s + 0.579·20-s + 0.0854·23-s + 0.341·25-s + 0.112·26-s + 0.0310·28-s − 1.57·29-s + 0.985·31-s − 0.176·32-s + 0.0718·35-s − 1.18·37-s + 0.374·38-s − 0.409·40-s − 0.0444·41-s − 0.958·43-s − 0.0604·46-s − 1.45·47-s − 0.996·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6534 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6534 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 11 | \( 1 \) |
good | 5 | \( 1 - 2.59T + 5T^{2} \) |
| 7 | \( 1 - 0.164T + 7T^{2} \) |
| 13 | \( 1 + 0.573T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 2.30T + 19T^{2} \) |
| 23 | \( 1 - 0.409T + 23T^{2} \) |
| 29 | \( 1 + 8.48T + 29T^{2} \) |
| 31 | \( 1 - 5.48T + 31T^{2} \) |
| 37 | \( 1 + 7.19T + 37T^{2} \) |
| 41 | \( 1 + 0.284T + 41T^{2} \) |
| 43 | \( 1 + 6.28T + 43T^{2} \) |
| 47 | \( 1 + 9.96T + 47T^{2} \) |
| 53 | \( 1 - 8.68T + 53T^{2} \) |
| 59 | \( 1 + 3.90T + 59T^{2} \) |
| 61 | \( 1 + 8.18T + 61T^{2} \) |
| 67 | \( 1 - 6.48T + 67T^{2} \) |
| 71 | \( 1 - 11.1T + 71T^{2} \) |
| 73 | \( 1 - 6.51T + 73T^{2} \) |
| 79 | \( 1 + 10.1T + 79T^{2} \) |
| 83 | \( 1 - 1.79T + 83T^{2} \) |
| 89 | \( 1 - 3.39T + 89T^{2} \) |
| 97 | \( 1 + 12.9T + 97T^{2} \) |
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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
−7.81779610151548947604760593049, −6.80244587555348113543917079617, −6.46809894985193297133366397191, −5.56681479239270162098705930069, −5.04405403645882546685501265235, −3.90763668038136625226924333541, −2.94402625893675780942320708055, −2.04348251601367723570963263291, −1.46206664875741545211366920868, 0,
1.46206664875741545211366920868, 2.04348251601367723570963263291, 2.94402625893675780942320708055, 3.90763668038136625226924333541, 5.04405403645882546685501265235, 5.56681479239270162098705930069, 6.46809894985193297133366397191, 6.80244587555348113543917079617, 7.81779610151548947604760593049