| L(s) = 1 | − 2.29·2-s + 1.42·3-s + 3.26·4-s + 0.129·5-s − 3.26·6-s + 5.26·7-s − 2.90·8-s − 0.970·9-s − 0.296·10-s − 11-s + 4.65·12-s + 3.51·13-s − 12.0·14-s + 0.184·15-s + 0.132·16-s − 0.540·17-s + 2.22·18-s − 0.334·19-s + 0.422·20-s + 7.50·21-s + 2.29·22-s − 0.904·23-s − 4.13·24-s − 4.98·25-s − 8.06·26-s − 5.65·27-s + 17.2·28-s + ⋯ |
| L(s) = 1 | − 1.62·2-s + 0.822·3-s + 1.63·4-s + 0.0578·5-s − 1.33·6-s + 1.99·7-s − 1.02·8-s − 0.323·9-s − 0.0938·10-s − 0.301·11-s + 1.34·12-s + 0.974·13-s − 3.23·14-s + 0.0475·15-s + 0.0330·16-s − 0.131·17-s + 0.524·18-s − 0.0766·19-s + 0.0944·20-s + 1.63·21-s + 0.489·22-s − 0.188·23-s − 0.844·24-s − 0.996·25-s − 1.58·26-s − 1.08·27-s + 3.25·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 341 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 341 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9709230384\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9709230384\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 + 2.29T + 2T^{2} \) |
| 3 | \( 1 - 1.42T + 3T^{2} \) |
| 5 | \( 1 - 0.129T + 5T^{2} \) |
| 7 | \( 1 - 5.26T + 7T^{2} \) |
| 13 | \( 1 - 3.51T + 13T^{2} \) |
| 17 | \( 1 + 0.540T + 17T^{2} \) |
| 19 | \( 1 + 0.334T + 19T^{2} \) |
| 23 | \( 1 + 0.904T + 23T^{2} \) |
| 29 | \( 1 - 6.91T + 29T^{2} \) |
| 37 | \( 1 - 5.97T + 37T^{2} \) |
| 41 | \( 1 + 0.490T + 41T^{2} \) |
| 43 | \( 1 - 2.57T + 43T^{2} \) |
| 47 | \( 1 - 5.55T + 47T^{2} \) |
| 53 | \( 1 - 1.08T + 53T^{2} \) |
| 59 | \( 1 + 9.04T + 59T^{2} \) |
| 61 | \( 1 + 7.98T + 61T^{2} \) |
| 67 | \( 1 + 6.94T + 67T^{2} \) |
| 71 | \( 1 - 14.0T + 71T^{2} \) |
| 73 | \( 1 + 2.35T + 73T^{2} \) |
| 79 | \( 1 + 12.2T + 79T^{2} \) |
| 83 | \( 1 + 10.3T + 83T^{2} \) |
| 89 | \( 1 + 10.8T + 89T^{2} \) |
| 97 | \( 1 + 0.0142T + 97T^{2} \) |
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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
−11.17290502650514108447368153387, −10.57136299190410424279703180022, −9.400214338361702095704860489234, −8.475064333529841326757594862020, −8.191950265121551463247036143640, −7.43286154406114048763249986488, −5.88812707906980104777215201158, −4.37825847503866865077171699056, −2.50861481026614334664281172769, −1.40997658461002008251187384492,
1.40997658461002008251187384492, 2.50861481026614334664281172769, 4.37825847503866865077171699056, 5.88812707906980104777215201158, 7.43286154406114048763249986488, 8.191950265121551463247036143640, 8.475064333529841326757594862020, 9.400214338361702095704860489234, 10.57136299190410424279703180022, 11.17290502650514108447368153387
