| L(s) = 1 | − 0.712·2-s + 3-s − 1.49·4-s − 0.712·6-s + 2.77·7-s + 2.48·8-s + 9-s + 4.26·11-s − 1.49·12-s − 0.779·13-s − 1.98·14-s + 1.21·16-s + 1.90·17-s − 0.712·18-s + 6.72·19-s + 2.77·21-s − 3.04·22-s − 2.17·23-s + 2.48·24-s + 0.555·26-s + 27-s − 4.14·28-s + 29-s − 8.82·31-s − 5.83·32-s + 4.26·33-s − 1.35·34-s + ⋯ |
| L(s) = 1 | − 0.503·2-s + 0.577·3-s − 0.746·4-s − 0.290·6-s + 1.05·7-s + 0.879·8-s + 0.333·9-s + 1.28·11-s − 0.430·12-s − 0.216·13-s − 0.529·14-s + 0.302·16-s + 0.461·17-s − 0.167·18-s + 1.54·19-s + 0.606·21-s − 0.648·22-s − 0.452·23-s + 0.507·24-s + 0.108·26-s + 0.192·27-s − 0.783·28-s + 0.185·29-s − 1.58·31-s − 1.03·32-s + 0.742·33-s − 0.232·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.830161120\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.830161120\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + 0.712T + 2T^{2} \) |
| 7 | \( 1 - 2.77T + 7T^{2} \) |
| 11 | \( 1 - 4.26T + 11T^{2} \) |
| 13 | \( 1 + 0.779T + 13T^{2} \) |
| 17 | \( 1 - 1.90T + 17T^{2} \) |
| 19 | \( 1 - 6.72T + 19T^{2} \) |
| 23 | \( 1 + 2.17T + 23T^{2} \) |
| 31 | \( 1 + 8.82T + 31T^{2} \) |
| 37 | \( 1 + 1.48T + 37T^{2} \) |
| 41 | \( 1 + 7.71T + 41T^{2} \) |
| 43 | \( 1 - 8.19T + 43T^{2} \) |
| 47 | \( 1 - 5.19T + 47T^{2} \) |
| 53 | \( 1 - 11.7T + 53T^{2} \) |
| 59 | \( 1 + 4.46T + 59T^{2} \) |
| 61 | \( 1 + 5.24T + 61T^{2} \) |
| 67 | \( 1 - 8.49T + 67T^{2} \) |
| 71 | \( 1 - 0.663T + 71T^{2} \) |
| 73 | \( 1 + 16.5T + 73T^{2} \) |
| 79 | \( 1 - 9.54T + 79T^{2} \) |
| 83 | \( 1 - 0.0123T + 83T^{2} \) |
| 89 | \( 1 + 5.46T + 89T^{2} \) |
| 97 | \( 1 - 0.952T + 97T^{2} \) |
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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
−9.009662271731103879223555134214, −8.470988116287035332471923740060, −7.57137773425025841365339471799, −7.21220420017377197627456492370, −5.77226877713427626041308812698, −4.99143255628435602660286254547, −4.12370012665830086078976485887, −3.40326561960749779188572113345, −1.86287764034738179322724801509, −1.04047428153701103143638325675,
1.04047428153701103143638325675, 1.86287764034738179322724801509, 3.40326561960749779188572113345, 4.12370012665830086078976485887, 4.99143255628435602660286254547, 5.77226877713427626041308812698, 7.21220420017377197627456492370, 7.57137773425025841365339471799, 8.470988116287035332471923740060, 9.009662271731103879223555134214
