L(s) = 1 | + 1.87·2-s − 3-s + 1.51·4-s + 0.415·5-s − 1.87·6-s + 5.01·7-s − 0.909·8-s + 9-s + 0.778·10-s − 1.51·12-s − 4.35·13-s + 9.39·14-s − 0.415·15-s − 4.73·16-s − 17-s + 1.87·18-s + 2.30·19-s + 0.628·20-s − 5.01·21-s − 6.55·23-s + 0.909·24-s − 4.82·25-s − 8.16·26-s − 27-s + 7.59·28-s − 7.38·29-s − 0.778·30-s + ⋯ |
L(s) = 1 | + 1.32·2-s − 0.577·3-s + 0.757·4-s + 0.185·5-s − 0.765·6-s + 1.89·7-s − 0.321·8-s + 0.333·9-s + 0.246·10-s − 0.437·12-s − 1.20·13-s + 2.51·14-s − 0.107·15-s − 1.18·16-s − 0.242·17-s + 0.441·18-s + 0.528·19-s + 0.140·20-s − 1.09·21-s − 1.36·23-s + 0.185·24-s − 0.965·25-s − 1.60·26-s − 0.192·27-s + 1.43·28-s − 1.37·29-s − 0.142·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| 17 | \( 1 + T \) |
good | 2 | \( 1 - 1.87T + 2T^{2} \) |
| 5 | \( 1 - 0.415T + 5T^{2} \) |
| 7 | \( 1 - 5.01T + 7T^{2} \) |
| 13 | \( 1 + 4.35T + 13T^{2} \) |
| 19 | \( 1 - 2.30T + 19T^{2} \) |
| 23 | \( 1 + 6.55T + 23T^{2} \) |
| 29 | \( 1 + 7.38T + 29T^{2} \) |
| 31 | \( 1 + 6.23T + 31T^{2} \) |
| 37 | \( 1 + 6.09T + 37T^{2} \) |
| 41 | \( 1 - 3.06T + 41T^{2} \) |
| 43 | \( 1 + 12.0T + 43T^{2} \) |
| 47 | \( 1 - 2.19T + 47T^{2} \) |
| 53 | \( 1 + 2.05T + 53T^{2} \) |
| 59 | \( 1 + 2.11T + 59T^{2} \) |
| 61 | \( 1 + 5.55T + 61T^{2} \) |
| 67 | \( 1 + 5.21T + 67T^{2} \) |
| 71 | \( 1 - 15.1T + 71T^{2} \) |
| 73 | \( 1 - 11.6T + 73T^{2} \) |
| 79 | \( 1 - 12.2T + 79T^{2} \) |
| 83 | \( 1 + 2.96T + 83T^{2} \) |
| 89 | \( 1 + 7.09T + 89T^{2} \) |
| 97 | \( 1 - 0.304T + 97T^{2} \) |
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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
−7.60065497737819605483033824830, −6.86959600353326936030486120868, −5.89815066331632752055956318438, −5.29294826846734550665306275593, −5.00499173230426974152903734739, −4.22066135163297090277990181188, −3.57933056448517990692148224812, −2.18441954194491518993483637018, −1.78140635094587115366211477532, 0,
1.78140635094587115366211477532, 2.18441954194491518993483637018, 3.57933056448517990692148224812, 4.22066135163297090277990181188, 5.00499173230426974152903734739, 5.29294826846734550665306275593, 5.89815066331632752055956318438, 6.86959600353326936030486120868, 7.60065497737819605483033824830