L(s) = 1 | + 1.30·2-s + 1.30·3-s − 0.302·4-s − 3.60·5-s + 1.69·6-s + 7-s − 3·8-s − 1.30·9-s − 4.69·10-s − 0.394·12-s + 0.605·13-s + 1.30·14-s − 4.69·15-s − 3.30·16-s − 6.30·17-s − 1.69·18-s + 3·19-s + 1.09·20-s + 1.30·21-s − 6.30·23-s − 3.90·24-s + 7.99·25-s + 0.788·26-s − 5.60·27-s − 0.302·28-s − 8.30·29-s − 6.11·30-s + ⋯ |
L(s) = 1 | + 0.921·2-s + 0.752·3-s − 0.151·4-s − 1.61·5-s + 0.692·6-s + 0.377·7-s − 1.06·8-s − 0.434·9-s − 1.48·10-s − 0.113·12-s + 0.167·13-s + 0.348·14-s − 1.21·15-s − 0.825·16-s − 1.52·17-s − 0.400·18-s + 0.688·19-s + 0.244·20-s + 0.284·21-s − 1.31·23-s − 0.797·24-s + 1.59·25-s + 0.154·26-s − 1.07·27-s − 0.0572·28-s − 1.54·29-s − 1.11·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - 1.30T + 2T^{2} \) |
| 3 | \( 1 - 1.30T + 3T^{2} \) |
| 5 | \( 1 + 3.60T + 5T^{2} \) |
| 13 | \( 1 - 0.605T + 13T^{2} \) |
| 17 | \( 1 + 6.30T + 17T^{2} \) |
| 19 | \( 1 - 3T + 19T^{2} \) |
| 23 | \( 1 + 6.30T + 23T^{2} \) |
| 29 | \( 1 + 8.30T + 29T^{2} \) |
| 31 | \( 1 - T + 31T^{2} \) |
| 37 | \( 1 - 9.21T + 37T^{2} \) |
| 41 | \( 1 + 7T + 41T^{2} \) |
| 43 | \( 1 + 5.30T + 43T^{2} \) |
| 47 | \( 1 - 8.90T + 47T^{2} \) |
| 53 | \( 1 + 2.09T + 53T^{2} \) |
| 59 | \( 1 - 10.3T + 59T^{2} \) |
| 61 | \( 1 - 0.697T + 61T^{2} \) |
| 67 | \( 1 + 9.51T + 67T^{2} \) |
| 71 | \( 1 + 0.697T + 71T^{2} \) |
| 73 | \( 1 - 5T + 73T^{2} \) |
| 79 | \( 1 - 4.69T + 79T^{2} \) |
| 83 | \( 1 + 3T + 83T^{2} \) |
| 89 | \( 1 - 17.7T + 89T^{2} \) |
| 97 | \( 1 - 3.60T + 97T^{2} \) |
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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
−9.521905598032116919222919751842, −8.712601211135035016907966407818, −8.142778635381997751974433290060, −7.35994214504320994676364845918, −6.13911453906827303303041972512, −5.02325427997661327658395067994, −4.05650803025711294069661877862, −3.62760808748633025766277171537, −2.46110966614524431889989442234, 0,
2.46110966614524431889989442234, 3.62760808748633025766277171537, 4.05650803025711294069661877862, 5.02325427997661327658395067994, 6.13911453906827303303041972512, 7.35994214504320994676364845918, 8.142778635381997751974433290060, 8.712601211135035016907966407818, 9.521905598032116919222919751842