L(s) = 1 | + 3-s − 2·4-s − 5-s + 7-s + 9-s + 3·11-s − 2·12-s − 4·13-s − 15-s + 4·16-s − 6·17-s − 19-s + 2·20-s + 21-s − 23-s + 25-s + 27-s − 2·28-s + 6·29-s + 8·31-s + 3·33-s − 35-s − 2·36-s − 10·37-s − 4·39-s − 3·41-s − 10·43-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 4-s − 0.447·5-s + 0.377·7-s + 1/3·9-s + 0.904·11-s − 0.577·12-s − 1.10·13-s − 0.258·15-s + 16-s − 1.45·17-s − 0.229·19-s + 0.447·20-s + 0.218·21-s − 0.208·23-s + 1/5·25-s + 0.192·27-s − 0.377·28-s + 1.11·29-s + 1.43·31-s + 0.522·33-s − 0.169·35-s − 1/3·36-s − 1.64·37-s − 0.640·39-s − 0.468·41-s − 1.52·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 2 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−8.485749227808954796631523822352, −8.160625928367124194564103725231, −7.04880326351232874256217328676, −6.44820708004920735214806528518, −5.00010326554341053545442709036, −4.58491344295942601284268345621, −3.81274100836837068759670803629, −2.78979023966689197511354892893, −1.54049047545509520664186580686, 0,
1.54049047545509520664186580686, 2.78979023966689197511354892893, 3.81274100836837068759670803629, 4.58491344295942601284268345621, 5.00010326554341053545442709036, 6.44820708004920735214806528518, 7.04880326351232874256217328676, 8.160625928367124194564103725231, 8.485749227808954796631523822352