| L(s) = 1 | + 2-s + 0.180·3-s + 4-s + 4.08·5-s + 0.180·6-s + 3.52·7-s + 8-s − 2.96·9-s + 4.08·10-s + 0.180·12-s + 1.08·13-s + 3.52·14-s + 0.739·15-s + 16-s − 3.90·17-s − 2.96·18-s − 19-s + 4.08·20-s + 0.638·21-s − 4.34·23-s + 0.180·24-s + 11.6·25-s + 1.08·26-s − 1.07·27-s + 3.52·28-s − 6.06·29-s + 0.739·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.104·3-s + 0.5·4-s + 1.82·5-s + 0.0738·6-s + 1.33·7-s + 0.353·8-s − 0.989·9-s + 1.29·10-s + 0.0522·12-s + 0.300·13-s + 0.942·14-s + 0.190·15-s + 0.250·16-s − 0.946·17-s − 0.699·18-s − 0.229·19-s + 0.913·20-s + 0.139·21-s − 0.905·23-s + 0.0369·24-s + 2.33·25-s + 0.212·26-s − 0.207·27-s + 0.666·28-s − 1.12·29-s + 0.134·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.094125009\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.094125009\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 11 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 0.180T + 3T^{2} \) |
| 5 | \( 1 - 4.08T + 5T^{2} \) |
| 7 | \( 1 - 3.52T + 7T^{2} \) |
| 13 | \( 1 - 1.08T + 13T^{2} \) |
| 17 | \( 1 + 3.90T + 17T^{2} \) |
| 23 | \( 1 + 4.34T + 23T^{2} \) |
| 29 | \( 1 + 6.06T + 29T^{2} \) |
| 31 | \( 1 - 7.26T + 31T^{2} \) |
| 37 | \( 1 - 0.754T + 37T^{2} \) |
| 41 | \( 1 - 6.67T + 41T^{2} \) |
| 43 | \( 1 - 4.60T + 43T^{2} \) |
| 47 | \( 1 - 12.1T + 47T^{2} \) |
| 53 | \( 1 - 4.42T + 53T^{2} \) |
| 59 | \( 1 + 3.54T + 59T^{2} \) |
| 61 | \( 1 - 9.05T + 61T^{2} \) |
| 67 | \( 1 - 1.35T + 67T^{2} \) |
| 71 | \( 1 + 8.96T + 71T^{2} \) |
| 73 | \( 1 - 8.26T + 73T^{2} \) |
| 79 | \( 1 - 11.5T + 79T^{2} \) |
| 83 | \( 1 + 17.6T + 83T^{2} \) |
| 89 | \( 1 - 12.2T + 89T^{2} \) |
| 97 | \( 1 - 16.4T + 97T^{2} \) |
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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
−8.391675058505327548888314549462, −7.57301809151525249170349887325, −6.55109782176740690206779876725, −5.91763715514310266135002897183, −5.50585801869072695852639885008, −4.74705619360292064878224705032, −3.92506277376722853497997564478, −2.40829878596657824325837787865, −2.35287060438431455140103728894, −1.23373707400815539812556394615,
1.23373707400815539812556394615, 2.35287060438431455140103728894, 2.40829878596657824325837787865, 3.92506277376722853497997564478, 4.74705619360292064878224705032, 5.50585801869072695852639885008, 5.91763715514310266135002897183, 6.55109782176740690206779876725, 7.57301809151525249170349887325, 8.391675058505327548888314549462
