| L(s) = 1 | − 3-s + 5-s + 3.47·7-s + 9-s + 6.27·11-s − 0.805·13-s − 15-s − 0.710·17-s − 3.47·21-s + 0.710·23-s + 25-s − 27-s + 7.51·29-s + 31-s − 6.27·33-s + 3.47·35-s + 5.47·37-s + 0.805·39-s − 4.27·41-s − 6.27·43-s + 45-s − 9.65·47-s + 5.04·49-s + 0.710·51-s − 11.6·53-s + 6.27·55-s − 13.8·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1.31·7-s + 0.333·9-s + 1.89·11-s − 0.223·13-s − 0.258·15-s − 0.172·17-s − 0.757·21-s + 0.148·23-s + 0.200·25-s − 0.192·27-s + 1.39·29-s + 0.179·31-s − 1.09·33-s + 0.586·35-s + 0.899·37-s + 0.129·39-s − 0.667·41-s − 0.957·43-s + 0.149·45-s − 1.40·47-s + 0.720·49-s + 0.0995·51-s − 1.60·53-s + 0.846·55-s − 1.80·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.091675780\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.091675780\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| good | 7 | \( 1 - 3.47T + 7T^{2} \) |
| 11 | \( 1 - 6.27T + 11T^{2} \) |
| 13 | \( 1 + 0.805T + 13T^{2} \) |
| 17 | \( 1 + 0.710T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 0.710T + 23T^{2} \) |
| 29 | \( 1 - 7.51T + 29T^{2} \) |
| 37 | \( 1 - 5.47T + 37T^{2} \) |
| 41 | \( 1 + 4.27T + 41T^{2} \) |
| 43 | \( 1 + 6.27T + 43T^{2} \) |
| 47 | \( 1 + 9.65T + 47T^{2} \) |
| 53 | \( 1 + 11.6T + 53T^{2} \) |
| 59 | \( 1 + 13.8T + 59T^{2} \) |
| 61 | \( 1 + 11.9T + 61T^{2} \) |
| 67 | \( 1 - 15.1T + 67T^{2} \) |
| 71 | \( 1 - 6.18T + 71T^{2} \) |
| 73 | \( 1 - 11.0T + 73T^{2} \) |
| 79 | \( 1 - 9.37T + 79T^{2} \) |
| 83 | \( 1 + 4.71T + 83T^{2} \) |
| 89 | \( 1 - 12.1T + 89T^{2} \) |
| 97 | \( 1 - 6.85T + 97T^{2} \) |
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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
−9.302631672976531951115595049774, −8.457351314420403446131325846918, −7.70981476805234378736411649925, −6.52144176354639406509458549930, −6.32042690351850996422216440465, −4.92451154106650328407147519166, −4.64704121024132259703949796097, −3.43299273482973327892985230932, −1.91022400832669488510696850939, −1.13971253702006093138842904092,
1.13971253702006093138842904092, 1.91022400832669488510696850939, 3.43299273482973327892985230932, 4.64704121024132259703949796097, 4.92451154106650328407147519166, 6.32042690351850996422216440465, 6.52144176354639406509458549930, 7.70981476805234378736411649925, 8.457351314420403446131325846918, 9.302631672976531951115595049774
