| L(s) = 1 | − 4.90·2-s − 3·3-s + 16.0·4-s + 5·5-s + 14.7·6-s + 0.166·7-s − 39.4·8-s + 9·9-s − 24.5·10-s − 48.1·12-s − 27.6·13-s − 0.816·14-s − 15·15-s + 65.0·16-s − 4.79·17-s − 44.1·18-s + 11.2·19-s + 80.2·20-s − 0.499·21-s + 107.·23-s + 118.·24-s + 25·25-s + 135.·26-s − 27·27-s + 2.67·28-s − 123.·29-s + 73.5·30-s + ⋯ |
| L(s) = 1 | − 1.73·2-s − 0.577·3-s + 2.00·4-s + 0.447·5-s + 1.00·6-s + 0.00899·7-s − 1.74·8-s + 0.333·9-s − 0.775·10-s − 1.15·12-s − 0.590·13-s − 0.0155·14-s − 0.258·15-s + 1.01·16-s − 0.0683·17-s − 0.577·18-s + 0.135·19-s + 0.896·20-s − 0.00519·21-s + 0.976·23-s + 1.00·24-s + 0.200·25-s + 1.02·26-s − 0.192·27-s + 0.0180·28-s − 0.787·29-s + 0.447·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 5 | \( 1 - 5T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 4.90T + 8T^{2} \) |
| 7 | \( 1 - 0.166T + 343T^{2} \) |
| 13 | \( 1 + 27.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 4.79T + 4.91e3T^{2} \) |
| 19 | \( 1 - 11.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 107.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 123.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 75.8T + 2.97e4T^{2} \) |
| 37 | \( 1 - 351.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 355.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 313.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 160.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 335.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 25.2T + 2.05e5T^{2} \) |
| 61 | \( 1 + 461.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 627.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 119.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 631.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 949.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 43.2T + 5.71e5T^{2} \) |
| 89 | \( 1 - 380.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.69e3T + 9.12e5T^{2} \) |
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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
−8.749952627156599155060500525365, −7.67435401858992274550526129993, −7.21616278541195617687422709214, −6.36622459791039458064541661099, −5.56485578845336320573200760227, −4.50733954676554717046116407081, −2.97224856955464870586848481319, −1.96165532761875106535495821422, −1.02568669114825941973407449074, 0,
1.02568669114825941973407449074, 1.96165532761875106535495821422, 2.97224856955464870586848481319, 4.50733954676554717046116407081, 5.56485578845336320573200760227, 6.36622459791039458064541661099, 7.21616278541195617687422709214, 7.67435401858992274550526129993, 8.749952627156599155060500525365
