| L(s) = 1 | + 0.430·2-s + 3-s − 1.81·4-s − 5-s + 0.430·6-s + 1.05·7-s − 1.64·8-s + 9-s − 0.430·10-s + 5.03·11-s − 1.81·12-s + 6.13·13-s + 0.452·14-s − 15-s + 2.92·16-s + 0.665·17-s + 0.430·18-s + 6.94·19-s + 1.81·20-s + 1.05·21-s + 2.16·22-s − 1.64·24-s + 25-s + 2.63·26-s + 27-s − 1.91·28-s − 5.59·29-s + ⋯ |
| L(s) = 1 | + 0.304·2-s + 0.577·3-s − 0.907·4-s − 0.447·5-s + 0.175·6-s + 0.397·7-s − 0.580·8-s + 0.333·9-s − 0.136·10-s + 1.51·11-s − 0.523·12-s + 1.70·13-s + 0.121·14-s − 0.258·15-s + 0.730·16-s + 0.161·17-s + 0.101·18-s + 1.59·19-s + 0.405·20-s + 0.229·21-s + 0.462·22-s − 0.335·24-s + 0.200·25-s + 0.517·26-s + 0.192·27-s − 0.360·28-s − 1.03·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.100380518\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.100380518\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 - 0.430T + 2T^{2} \) |
| 7 | \( 1 - 1.05T + 7T^{2} \) |
| 11 | \( 1 - 5.03T + 11T^{2} \) |
| 13 | \( 1 - 6.13T + 13T^{2} \) |
| 17 | \( 1 - 0.665T + 17T^{2} \) |
| 19 | \( 1 - 6.94T + 19T^{2} \) |
| 29 | \( 1 + 5.59T + 29T^{2} \) |
| 31 | \( 1 - 10.4T + 31T^{2} \) |
| 37 | \( 1 + 2.93T + 37T^{2} \) |
| 41 | \( 1 - 5.78T + 41T^{2} \) |
| 43 | \( 1 + 11.2T + 43T^{2} \) |
| 47 | \( 1 - 1.07T + 47T^{2} \) |
| 53 | \( 1 - 12.4T + 53T^{2} \) |
| 59 | \( 1 + 5.52T + 59T^{2} \) |
| 61 | \( 1 + 0.649T + 61T^{2} \) |
| 67 | \( 1 + 9.01T + 67T^{2} \) |
| 71 | \( 1 + 3.99T + 71T^{2} \) |
| 73 | \( 1 + 11.6T + 73T^{2} \) |
| 79 | \( 1 + 11.1T + 79T^{2} \) |
| 83 | \( 1 - 14.6T + 83T^{2} \) |
| 89 | \( 1 - 11.9T + 89T^{2} \) |
| 97 | \( 1 - 1.28T + 97T^{2} \) |
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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
−7.968432733303841877747069416677, −7.26828292508992830642382107569, −6.34950782584421145053181715817, −5.77343456250625014395302637656, −4.82688612202817441139678497868, −4.17936356158913375917703406233, −3.55515259204445319435866779709, −3.12629278985853321215922463717, −1.51826231015634110915916497545, −0.940974129781193425715853097682,
0.940974129781193425715853097682, 1.51826231015634110915916497545, 3.12629278985853321215922463717, 3.55515259204445319435866779709, 4.17936356158913375917703406233, 4.82688612202817441139678497868, 5.77343456250625014395302637656, 6.34950782584421145053181715817, 7.26828292508992830642382107569, 7.968432733303841877747069416677
