| L(s) = 1 | − 1.51·2-s − 3-s + 0.300·4-s − 5-s + 1.51·6-s + 3.34·7-s + 2.57·8-s + 9-s + 1.51·10-s − 1.18·11-s − 0.300·12-s + 0.915·13-s − 5.06·14-s + 15-s − 4.51·16-s + 2.52·17-s − 1.51·18-s + 8.46·19-s − 0.300·20-s − 3.34·21-s + 1.79·22-s − 2.57·24-s + 25-s − 1.38·26-s − 27-s + 1.00·28-s + 7.86·29-s + ⋯ |
| L(s) = 1 | − 1.07·2-s − 0.577·3-s + 0.150·4-s − 0.447·5-s + 0.619·6-s + 1.26·7-s + 0.911·8-s + 0.333·9-s + 0.479·10-s − 0.357·11-s − 0.0868·12-s + 0.253·13-s − 1.35·14-s + 0.258·15-s − 1.12·16-s + 0.611·17-s − 0.357·18-s + 1.94·19-s − 0.0672·20-s − 0.729·21-s + 0.383·22-s − 0.526·24-s + 0.200·25-s − 0.272·26-s − 0.192·27-s + 0.190·28-s + 1.46·29-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
| \(L(1)\) |
\(\approx\) |
\(1.129059203\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.129059203\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.51T + 2T^{2} \) |
| 7 | \( 1 - 3.34T + 7T^{2} \) |
| 11 | \( 1 + 1.18T + 11T^{2} \) |
| 13 | \( 1 - 0.915T + 13T^{2} \) |
| 17 | \( 1 - 2.52T + 17T^{2} \) |
| 19 | \( 1 - 8.46T + 19T^{2} \) |
| 29 | \( 1 - 7.86T + 29T^{2} \) |
| 31 | \( 1 + 5.67T + 31T^{2} \) |
| 37 | \( 1 - 5.68T + 37T^{2} \) |
| 41 | \( 1 - 10.1T + 41T^{2} \) |
| 43 | \( 1 - 8.92T + 43T^{2} \) |
| 47 | \( 1 + 7.20T + 47T^{2} \) |
| 53 | \( 1 - 1.10T + 53T^{2} \) |
| 59 | \( 1 + 3.44T + 59T^{2} \) |
| 61 | \( 1 - 4.49T + 61T^{2} \) |
| 67 | \( 1 + 0.818T + 67T^{2} \) |
| 71 | \( 1 - 7.34T + 71T^{2} \) |
| 73 | \( 1 - 13.6T + 73T^{2} \) |
| 79 | \( 1 - 15.6T + 79T^{2} \) |
| 83 | \( 1 - 6.60T + 83T^{2} \) |
| 89 | \( 1 + 9.17T + 89T^{2} \) |
| 97 | \( 1 + 3.08T + 97T^{2} \) |
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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
−7.77320709094556005261715337970, −7.61457939227482523965953033082, −6.69640979481340452756137073952, −5.62512297468150036692913498217, −5.06200728251618247721120075520, −4.47631423548008679154768327739, −3.58240931678492136494696967378, −2.39569394917528622815939424370, −1.22736638663122690390125125151, −0.794909537145443682647944264843,
0.794909537145443682647944264843, 1.22736638663122690390125125151, 2.39569394917528622815939424370, 3.58240931678492136494696967378, 4.47631423548008679154768327739, 5.06200728251618247721120075520, 5.62512297468150036692913498217, 6.69640979481340452756137073952, 7.61457939227482523965953033082, 7.77320709094556005261715337970
