| L(s) = 1 | − 2.23·2-s + 2.98·4-s − 1.03·7-s − 2.20·8-s − 6.17·11-s + 0.937·13-s + 2.30·14-s − 1.04·16-s + 6.56·17-s + 5.67·19-s + 13.7·22-s − 1.64·23-s − 2.09·26-s − 3.08·28-s − 8.35·29-s + 5.53·31-s + 6.74·32-s − 14.6·34-s − 1.29·37-s − 12.6·38-s + 4.98·41-s + 7.75·43-s − 18.4·44-s + 3.67·46-s + 7.67·47-s − 5.93·49-s + 2.80·52-s + ⋯ |
| L(s) = 1 | − 1.57·2-s + 1.49·4-s − 0.389·7-s − 0.781·8-s − 1.86·11-s + 0.260·13-s + 0.615·14-s − 0.260·16-s + 1.59·17-s + 1.30·19-s + 2.94·22-s − 0.343·23-s − 0.410·26-s − 0.582·28-s − 1.55·29-s + 0.993·31-s + 1.19·32-s − 2.51·34-s − 0.212·37-s − 2.05·38-s + 0.777·41-s + 1.18·43-s − 2.78·44-s + 0.542·46-s + 1.11·47-s − 0.848·49-s + 0.388·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6151604690\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6151604690\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 2.23T + 2T^{2} \) |
| 7 | \( 1 + 1.03T + 7T^{2} \) |
| 11 | \( 1 + 6.17T + 11T^{2} \) |
| 13 | \( 1 - 0.937T + 13T^{2} \) |
| 17 | \( 1 - 6.56T + 17T^{2} \) |
| 19 | \( 1 - 5.67T + 19T^{2} \) |
| 23 | \( 1 + 1.64T + 23T^{2} \) |
| 29 | \( 1 + 8.35T + 29T^{2} \) |
| 31 | \( 1 - 5.53T + 31T^{2} \) |
| 37 | \( 1 + 1.29T + 37T^{2} \) |
| 41 | \( 1 - 4.98T + 41T^{2} \) |
| 43 | \( 1 - 7.75T + 43T^{2} \) |
| 47 | \( 1 - 7.67T + 47T^{2} \) |
| 53 | \( 1 + 0.500T + 53T^{2} \) |
| 59 | \( 1 - 1.19T + 59T^{2} \) |
| 61 | \( 1 + 12.3T + 61T^{2} \) |
| 67 | \( 1 + 7.58T + 67T^{2} \) |
| 71 | \( 1 + 10.6T + 71T^{2} \) |
| 73 | \( 1 + 7.98T + 73T^{2} \) |
| 79 | \( 1 + 13.9T + 79T^{2} \) |
| 83 | \( 1 + 1.46T + 83T^{2} \) |
| 89 | \( 1 - 8.51T + 89T^{2} \) |
| 97 | \( 1 - 3.75T + 97T^{2} \) |
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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
−8.001616092699540360657197006172, −7.56615048201996409372213350979, −7.31441402802714000191742535413, −5.95089185897175474506215308821, −5.60249064986561245060111291271, −4.54163733288515163408928607564, −3.24747273706677750198807491255, −2.67394725384526081579066110512, −1.54717968734540012293713139736, −0.54546863885044430493204918669,
0.54546863885044430493204918669, 1.54717968734540012293713139736, 2.67394725384526081579066110512, 3.24747273706677750198807491255, 4.54163733288515163408928607564, 5.60249064986561245060111291271, 5.95089185897175474506215308821, 7.31441402802714000191742535413, 7.56615048201996409372213350979, 8.001616092699540360657197006172
