| L(s) = 1 | − 2-s + 4-s + 3.23·5-s + 4.23·7-s − 8-s − 3.23·10-s − 1.62·11-s + 0.843·13-s − 4.23·14-s + 16-s + 0.942·17-s − 4.16·19-s + 3.23·20-s + 1.62·22-s + 1.50·23-s + 5.45·25-s − 0.843·26-s + 4.23·28-s − 9.60·29-s − 3.22·31-s − 32-s − 0.942·34-s + 13.7·35-s − 0.461·37-s + 4.16·38-s − 3.23·40-s + 12.3·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.44·5-s + 1.60·7-s − 0.353·8-s − 1.02·10-s − 0.490·11-s + 0.234·13-s − 1.13·14-s + 0.250·16-s + 0.228·17-s − 0.955·19-s + 0.722·20-s + 0.346·22-s + 0.314·23-s + 1.09·25-s − 0.165·26-s + 0.800·28-s − 1.78·29-s − 0.580·31-s − 0.176·32-s − 0.161·34-s + 2.31·35-s − 0.0758·37-s + 0.675·38-s − 0.511·40-s + 1.93·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.248046729\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.248046729\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 223 | \( 1 + T \) |
| good | 5 | \( 1 - 3.23T + 5T^{2} \) |
| 7 | \( 1 - 4.23T + 7T^{2} \) |
| 11 | \( 1 + 1.62T + 11T^{2} \) |
| 13 | \( 1 - 0.843T + 13T^{2} \) |
| 17 | \( 1 - 0.942T + 17T^{2} \) |
| 19 | \( 1 + 4.16T + 19T^{2} \) |
| 23 | \( 1 - 1.50T + 23T^{2} \) |
| 29 | \( 1 + 9.60T + 29T^{2} \) |
| 31 | \( 1 + 3.22T + 31T^{2} \) |
| 37 | \( 1 + 0.461T + 37T^{2} \) |
| 41 | \( 1 - 12.3T + 41T^{2} \) |
| 43 | \( 1 - 2.06T + 43T^{2} \) |
| 47 | \( 1 - 8.12T + 47T^{2} \) |
| 53 | \( 1 - 7.87T + 53T^{2} \) |
| 59 | \( 1 - 3.22T + 59T^{2} \) |
| 61 | \( 1 + 1.92T + 61T^{2} \) |
| 67 | \( 1 - 3.16T + 67T^{2} \) |
| 71 | \( 1 - 6.48T + 71T^{2} \) |
| 73 | \( 1 - 5.90T + 73T^{2} \) |
| 79 | \( 1 - 5.22T + 79T^{2} \) |
| 83 | \( 1 + 9.91T + 83T^{2} \) |
| 89 | \( 1 - 18.3T + 89T^{2} \) |
| 97 | \( 1 - 14.2T + 97T^{2} \) |
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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
−8.584246969564475327403373221601, −7.72774855543749123635288422595, −7.23461905561797060938242088246, −6.08505528758327964294622819950, −5.62136190720790329326721789005, −4.91196874059934294524427234799, −3.85378975391891721994702106401, −2.32323267076984549157249059962, −2.04062212049707104656044750934, −1.00524815863611790209033080791,
1.00524815863611790209033080791, 2.04062212049707104656044750934, 2.32323267076984549157249059962, 3.85378975391891721994702106401, 4.91196874059934294524427234799, 5.62136190720790329326721789005, 6.08505528758327964294622819950, 7.23461905561797060938242088246, 7.72774855543749123635288422595, 8.584246969564475327403373221601
