| L(s) = 1 | + 2.46·2-s + 4.05·4-s − 2.59·5-s + 5.05·8-s − 6.38·10-s − 4.51·11-s − 13-s + 4.32·16-s − 0.945·17-s + 4.05·19-s − 10.5·20-s − 11.1·22-s + 0.273·23-s + 1.72·25-s − 2.46·26-s − 2.46·29-s − 2.32·31-s + 0.539·32-s − 2.32·34-s + 1.78·37-s + 9.97·38-s − 13.1·40-s − 6.40·41-s − 10.4·43-s − 18.3·44-s + 0.672·46-s − 12.1·47-s + ⋯ |
| L(s) = 1 | + 1.73·2-s + 2.02·4-s − 1.15·5-s + 1.78·8-s − 2.01·10-s − 1.36·11-s − 0.277·13-s + 1.08·16-s − 0.229·17-s + 0.930·19-s − 2.35·20-s − 2.36·22-s + 0.0569·23-s + 0.345·25-s − 0.482·26-s − 0.456·29-s − 0.418·31-s + 0.0953·32-s − 0.399·34-s + 0.292·37-s + 1.61·38-s − 2.07·40-s − 1.00·41-s − 1.59·43-s − 2.75·44-s + 0.0991·46-s − 1.77·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 2.46T + 2T^{2} \) |
| 5 | \( 1 + 2.59T + 5T^{2} \) |
| 11 | \( 1 + 4.51T + 11T^{2} \) |
| 13 | \( 1 + T + 13T^{2} \) |
| 17 | \( 1 + 0.945T + 17T^{2} \) |
| 19 | \( 1 - 4.05T + 19T^{2} \) |
| 23 | \( 1 - 0.273T + 23T^{2} \) |
| 29 | \( 1 + 2.46T + 29T^{2} \) |
| 31 | \( 1 + 2.32T + 31T^{2} \) |
| 37 | \( 1 - 1.78T + 37T^{2} \) |
| 41 | \( 1 + 6.40T + 41T^{2} \) |
| 43 | \( 1 + 10.4T + 43T^{2} \) |
| 47 | \( 1 + 12.1T + 47T^{2} \) |
| 53 | \( 1 - 6.27T + 53T^{2} \) |
| 59 | \( 1 + 2.72T + 59T^{2} \) |
| 61 | \( 1 - 2.27T + 61T^{2} \) |
| 67 | \( 1 + 15.8T + 67T^{2} \) |
| 71 | \( 1 + 3.27T + 71T^{2} \) |
| 73 | \( 1 - 1.50T + 73T^{2} \) |
| 79 | \( 1 - 14.7T + 79T^{2} \) |
| 83 | \( 1 + 0.945T + 83T^{2} \) |
| 89 | \( 1 + 14.3T + 89T^{2} \) |
| 97 | \( 1 - 11.4T + 97T^{2} \) |
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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
−7.76377787061621477000513549092, −7.27574662841533374055707962885, −6.51089650139329965803616046752, −5.50890611876130950840844187885, −5.02398560038723280490574348793, −4.35091696646851235085397996437, −3.42690096330913641133670655861, −3.00931333272253712871700527157, −1.89705827130811851980395696501, 0,
1.89705827130811851980395696501, 3.00931333272253712871700527157, 3.42690096330913641133670655861, 4.35091696646851235085397996437, 5.02398560038723280490574348793, 5.50890611876130950840844187885, 6.51089650139329965803616046752, 7.27574662841533374055707962885, 7.76377787061621477000513549092
