L(s) = 1 | − 2·2-s + 3-s + 2·4-s + 5-s − 2·6-s + 7-s + 9-s − 2·10-s + 11-s + 2·12-s − 6·13-s − 2·14-s + 15-s − 4·16-s − 7·17-s − 2·18-s − 5·19-s + 2·20-s + 21-s − 2·22-s − 23-s + 25-s + 12·26-s + 27-s + 2·28-s − 5·29-s − 2·30-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 0.577·3-s + 4-s + 0.447·5-s − 0.816·6-s + 0.377·7-s + 1/3·9-s − 0.632·10-s + 0.301·11-s + 0.577·12-s − 1.66·13-s − 0.534·14-s + 0.258·15-s − 16-s − 1.69·17-s − 0.471·18-s − 1.14·19-s + 0.447·20-s + 0.218·21-s − 0.426·22-s − 0.208·23-s + 1/5·25-s + 2.35·26-s + 0.192·27-s + 0.377·28-s − 0.928·29-s − 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
good | 2 | \( 1 + p T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 - 3 T + p T^{2} \) |
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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
−9.212018498778742248955877369839, −8.846741941824141489575810143074, −7.87055662804250090687136539993, −7.22111789896158412118454288803, −6.44495083083266857141010156107, −5.00030635265137986421053214439, −4.12790788870104437563686789164, −2.36959749735691481963089848636, −1.87472727289654385991039771085, 0,
1.87472727289654385991039771085, 2.36959749735691481963089848636, 4.12790788870104437563686789164, 5.00030635265137986421053214439, 6.44495083083266857141010156107, 7.22111789896158412118454288803, 7.87055662804250090687136539993, 8.846741941824141489575810143074, 9.212018498778742248955877369839