L(s) = 1 | + 2-s − 3.17·3-s + 4-s + 2.94·5-s − 3.17·6-s − 2.37·7-s + 8-s + 7.10·9-s + 2.94·10-s + 0.0808·11-s − 3.17·12-s + 4.21·13-s − 2.37·14-s − 9.34·15-s + 16-s − 1.48·17-s + 7.10·18-s − 5.97·19-s + 2.94·20-s + 7.53·21-s + 0.0808·22-s − 7.06·23-s − 3.17·24-s + 3.65·25-s + 4.21·26-s − 13.0·27-s − 2.37·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.83·3-s + 0.5·4-s + 1.31·5-s − 1.29·6-s − 0.896·7-s + 0.353·8-s + 2.36·9-s + 0.930·10-s + 0.0243·11-s − 0.917·12-s + 1.16·13-s − 0.633·14-s − 2.41·15-s + 0.250·16-s − 0.360·17-s + 1.67·18-s − 1.36·19-s + 0.657·20-s + 1.64·21-s + 0.0172·22-s − 1.47·23-s − 0.648·24-s + 0.730·25-s + 0.826·26-s − 2.51·27-s − 0.448·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 2011 | \( 1 - T \) |
good | 3 | \( 1 + 3.17T + 3T^{2} \) |
| 5 | \( 1 - 2.94T + 5T^{2} \) |
| 7 | \( 1 + 2.37T + 7T^{2} \) |
| 11 | \( 1 - 0.0808T + 11T^{2} \) |
| 13 | \( 1 - 4.21T + 13T^{2} \) |
| 17 | \( 1 + 1.48T + 17T^{2} \) |
| 19 | \( 1 + 5.97T + 19T^{2} \) |
| 23 | \( 1 + 7.06T + 23T^{2} \) |
| 29 | \( 1 + 6.65T + 29T^{2} \) |
| 31 | \( 1 + 4.37T + 31T^{2} \) |
| 37 | \( 1 + 3.06T + 37T^{2} \) |
| 41 | \( 1 - 8.70T + 41T^{2} \) |
| 43 | \( 1 - 8.55T + 43T^{2} \) |
| 47 | \( 1 - 4.15T + 47T^{2} \) |
| 53 | \( 1 + 9.62T + 53T^{2} \) |
| 59 | \( 1 - 3.70T + 59T^{2} \) |
| 61 | \( 1 + 5.39T + 61T^{2} \) |
| 67 | \( 1 - 5.55T + 67T^{2} \) |
| 71 | \( 1 + 0.130T + 71T^{2} \) |
| 73 | \( 1 + 4.04T + 73T^{2} \) |
| 79 | \( 1 + 0.158T + 79T^{2} \) |
| 83 | \( 1 - 3.40T + 83T^{2} \) |
| 89 | \( 1 - 7.82T + 89T^{2} \) |
| 97 | \( 1 + 16.9T + 97T^{2} \) |
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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
−7.78728225507779006890792831366, −6.71173282238894752690304293922, −6.34983790603371301891347360522, −5.81003675297767006232452266521, −5.50815758793354967668166218089, −4.33093835510197425034088137859, −3.80116628252133017806569939815, −2.28273146413985345742536477706, −1.46264698511180180411148194177, 0,
1.46264698511180180411148194177, 2.28273146413985345742536477706, 3.80116628252133017806569939815, 4.33093835510197425034088137859, 5.50815758793354967668166218089, 5.81003675297767006232452266521, 6.34983790603371301891347360522, 6.71173282238894752690304293922, 7.78728225507779006890792831366