| L(s) = 1 | + 2.54·2-s + 3-s + 4.45·4-s + 5-s + 2.54·6-s + 1.14·7-s + 6.23·8-s + 9-s + 2.54·10-s + 3.34·11-s + 4.45·12-s − 2.70·13-s + 2.90·14-s + 15-s + 6.94·16-s + 6.70·17-s + 2.54·18-s + 1.16·19-s + 4.45·20-s + 1.14·21-s + 8.50·22-s + 6.23·24-s + 25-s − 6.87·26-s + 27-s + 5.09·28-s − 4.73·29-s + ⋯ |
| L(s) = 1 | + 1.79·2-s + 0.577·3-s + 2.22·4-s + 0.447·5-s + 1.03·6-s + 0.431·7-s + 2.20·8-s + 0.333·9-s + 0.803·10-s + 1.00·11-s + 1.28·12-s − 0.750·13-s + 0.775·14-s + 0.258·15-s + 1.73·16-s + 1.62·17-s + 0.598·18-s + 0.267·19-s + 0.996·20-s + 0.249·21-s + 1.81·22-s + 1.27·24-s + 0.200·25-s − 1.34·26-s + 0.192·27-s + 0.962·28-s − 0.878·29-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
| \(L(1)\) |
\(\approx\) |
\(10.37860457\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.37860457\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.54T + 2T^{2} \) |
| 7 | \( 1 - 1.14T + 7T^{2} \) |
| 11 | \( 1 - 3.34T + 11T^{2} \) |
| 13 | \( 1 + 2.70T + 13T^{2} \) |
| 17 | \( 1 - 6.70T + 17T^{2} \) |
| 19 | \( 1 - 1.16T + 19T^{2} \) |
| 29 | \( 1 + 4.73T + 29T^{2} \) |
| 31 | \( 1 + 9.58T + 31T^{2} \) |
| 37 | \( 1 + 7.96T + 37T^{2} \) |
| 41 | \( 1 - 3.24T + 41T^{2} \) |
| 43 | \( 1 - 11.8T + 43T^{2} \) |
| 47 | \( 1 + 1.14T + 47T^{2} \) |
| 53 | \( 1 + 6.62T + 53T^{2} \) |
| 59 | \( 1 - 4.14T + 59T^{2} \) |
| 61 | \( 1 - 12.2T + 61T^{2} \) |
| 67 | \( 1 + 3.54T + 67T^{2} \) |
| 71 | \( 1 - 2.64T + 71T^{2} \) |
| 73 | \( 1 - 14.2T + 73T^{2} \) |
| 79 | \( 1 - 4.83T + 79T^{2} \) |
| 83 | \( 1 + 15.3T + 83T^{2} \) |
| 89 | \( 1 + 5.33T + 89T^{2} \) |
| 97 | \( 1 + 5.42T + 97T^{2} \) |
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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
−7.44718670765281330581351125610, −7.11736910969015627746299129726, −6.26389517737541914270658055661, −5.37832497279607360707995061931, −5.25715702654105187753284812057, −4.11016499575165402398830746242, −3.68898996662256778521009912253, −2.93282471179533297712544031172, −2.05841590858901394270843295606, −1.37603028750643096936766698279,
1.37603028750643096936766698279, 2.05841590858901394270843295606, 2.93282471179533297712544031172, 3.68898996662256778521009912253, 4.11016499575165402398830746242, 5.25715702654105187753284812057, 5.37832497279607360707995061931, 6.26389517737541914270658055661, 7.11736910969015627746299129726, 7.44718670765281330581351125610
