| L(s) = 1 | − 1.24·2-s − 3-s − 0.445·4-s − 5-s + 1.24·6-s − 2.24·7-s + 3.04·8-s + 9-s + 1.24·10-s + 11-s + 0.445·12-s + 2.80·14-s + 15-s − 2.91·16-s − 6.49·17-s − 1.24·18-s + 2.33·19-s + 0.445·20-s + 2.24·21-s − 1.24·22-s + 0.198·23-s − 3.04·24-s + 25-s − 27-s + 1.00·28-s + 3.82·29-s − 1.24·30-s + ⋯ |
| L(s) = 1 | − 0.881·2-s − 0.577·3-s − 0.222·4-s − 0.447·5-s + 0.509·6-s − 0.849·7-s + 1.07·8-s + 0.333·9-s + 0.394·10-s + 0.301·11-s + 0.128·12-s + 0.748·14-s + 0.258·15-s − 0.727·16-s − 1.57·17-s − 0.293·18-s + 0.535·19-s + 0.0995·20-s + 0.490·21-s − 0.265·22-s + 0.0412·23-s − 0.622·24-s + 0.200·25-s − 0.192·27-s + 0.188·28-s + 0.711·29-s − 0.227·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 1.24T + 2T^{2} \) |
| 7 | \( 1 + 2.24T + 7T^{2} \) |
| 11 | \( 1 - T + 11T^{2} \) |
| 17 | \( 1 + 6.49T + 17T^{2} \) |
| 19 | \( 1 - 2.33T + 19T^{2} \) |
| 23 | \( 1 - 0.198T + 23T^{2} \) |
| 29 | \( 1 - 3.82T + 29T^{2} \) |
| 31 | \( 1 - 1.71T + 31T^{2} \) |
| 37 | \( 1 - 4.54T + 37T^{2} \) |
| 41 | \( 1 - 5.75T + 41T^{2} \) |
| 43 | \( 1 + 3.75T + 43T^{2} \) |
| 47 | \( 1 + 1.97T + 47T^{2} \) |
| 53 | \( 1 - 12.0T + 53T^{2} \) |
| 59 | \( 1 - 9.83T + 59T^{2} \) |
| 61 | \( 1 + 13.6T + 61T^{2} \) |
| 67 | \( 1 - 7.91T + 67T^{2} \) |
| 71 | \( 1 - 7.14T + 71T^{2} \) |
| 73 | \( 1 - 14.3T + 73T^{2} \) |
| 79 | \( 1 - 10.6T + 79T^{2} \) |
| 83 | \( 1 + 12.7T + 83T^{2} \) |
| 89 | \( 1 + 6.87T + 89T^{2} \) |
| 97 | \( 1 + 11.1T + 97T^{2} \) |
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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
−8.573763659660750124195057799766, −7.917649752351751080374659733540, −6.90918201907234834414354370918, −6.55243767521794765158675291987, −5.38189415468360140127917521028, −4.48035798694584552584031519487, −3.82015412241271114323695408498, −2.49752667670792693968300057596, −1.04932854406249215648068485345, 0,
1.04932854406249215648068485345, 2.49752667670792693968300057596, 3.82015412241271114323695408498, 4.48035798694584552584031519487, 5.38189415468360140127917521028, 6.55243767521794765158675291987, 6.90918201907234834414354370918, 7.917649752351751080374659733540, 8.573763659660750124195057799766
