| L(s) = 1 | − 2.48·2-s + 4.19·4-s − 5-s + 1.19·7-s − 5.46·8-s + 2.48·10-s − 1.19·11-s − 2.97·14-s + 5.21·16-s − 6.17·17-s − 6.97·19-s − 4.19·20-s + 2.97·22-s − 4.17·23-s + 25-s + 5.02·28-s − 6·29-s + 2.97·31-s − 2.05·32-s + 15.3·34-s − 1.19·35-s − 7.78·37-s + 17.3·38-s + 5.46·40-s − 6.17·41-s − 9.95·43-s − 5.02·44-s + ⋯ |
| L(s) = 1 | − 1.76·2-s + 2.09·4-s − 0.447·5-s + 0.452·7-s − 1.93·8-s + 0.787·10-s − 0.360·11-s − 0.796·14-s + 1.30·16-s − 1.49·17-s − 1.60·19-s − 0.938·20-s + 0.635·22-s − 0.870·23-s + 0.200·25-s + 0.948·28-s − 1.11·29-s + 0.534·31-s − 0.362·32-s + 2.63·34-s − 0.202·35-s − 1.27·37-s + 2.81·38-s + 0.864·40-s − 0.964·41-s − 1.51·43-s − 0.757·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2086245297\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2086245297\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 2.48T + 2T^{2} \) |
| 7 | \( 1 - 1.19T + 7T^{2} \) |
| 11 | \( 1 + 1.19T + 11T^{2} \) |
| 17 | \( 1 + 6.17T + 17T^{2} \) |
| 19 | \( 1 + 6.97T + 19T^{2} \) |
| 23 | \( 1 + 4.17T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 2.97T + 31T^{2} \) |
| 37 | \( 1 + 7.78T + 37T^{2} \) |
| 41 | \( 1 + 6.17T + 41T^{2} \) |
| 43 | \( 1 + 9.95T + 43T^{2} \) |
| 47 | \( 1 + 1.02T + 47T^{2} \) |
| 53 | \( 1 + 10.1T + 53T^{2} \) |
| 59 | \( 1 - 5.37T + 59T^{2} \) |
| 61 | \( 1 - 12.5T + 61T^{2} \) |
| 67 | \( 1 + 9.37T + 67T^{2} \) |
| 71 | \( 1 + 5.19T + 71T^{2} \) |
| 73 | \( 1 - 11.9T + 73T^{2} \) |
| 79 | \( 1 + 1.78T + 79T^{2} \) |
| 83 | \( 1 + 5.37T + 83T^{2} \) |
| 89 | \( 1 - 10.1T + 89T^{2} \) |
| 97 | \( 1 - 1.82T + 97T^{2} \) |
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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
−8.186836190204376217195969961344, −7.39720045259168191916278058065, −6.68742209196189899712721193103, −6.28171295583257598830389680491, −5.07085030556174251929790487843, −4.30683406640865254030282928184, −3.30236901786848724988133736313, −2.07772235932025980745268000985, −1.83378156851657311104279269259, −0.28343701912383238950952337329,
0.28343701912383238950952337329, 1.83378156851657311104279269259, 2.07772235932025980745268000985, 3.30236901786848724988133736313, 4.30683406640865254030282928184, 5.07085030556174251929790487843, 6.28171295583257598830389680491, 6.68742209196189899712721193103, 7.39720045259168191916278058065, 8.186836190204376217195969961344
