| L(s) = 1 | − 2.27·2-s + 3-s + 3.16·4-s − 4.15·5-s − 2.27·6-s − 2.64·8-s + 9-s + 9.43·10-s − 3.18·11-s + 3.16·12-s − 0.175·13-s − 4.15·15-s − 0.312·16-s + 1.32·17-s − 2.27·18-s − 0.0504·19-s − 13.1·20-s + 7.24·22-s − 23-s − 2.64·24-s + 12.2·25-s + 0.399·26-s + 27-s + 1.10·29-s + 9.43·30-s + 8.71·31-s + 6.00·32-s + ⋯ |
| L(s) = 1 | − 1.60·2-s + 0.577·3-s + 1.58·4-s − 1.85·5-s − 0.927·6-s − 0.936·8-s + 0.333·9-s + 2.98·10-s − 0.960·11-s + 0.913·12-s − 0.0487·13-s − 1.07·15-s − 0.0781·16-s + 0.321·17-s − 0.535·18-s − 0.0115·19-s − 2.93·20-s + 1.54·22-s − 0.208·23-s − 0.540·24-s + 2.44·25-s + 0.0782·26-s + 0.192·27-s + 0.204·29-s + 1.72·30-s + 1.56·31-s + 1.06·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 + 2.27T + 2T^{2} \) |
| 5 | \( 1 + 4.15T + 5T^{2} \) |
| 11 | \( 1 + 3.18T + 11T^{2} \) |
| 13 | \( 1 + 0.175T + 13T^{2} \) |
| 17 | \( 1 - 1.32T + 17T^{2} \) |
| 19 | \( 1 + 0.0504T + 19T^{2} \) |
| 29 | \( 1 - 1.10T + 29T^{2} \) |
| 31 | \( 1 - 8.71T + 31T^{2} \) |
| 37 | \( 1 - 3.76T + 37T^{2} \) |
| 41 | \( 1 + 4.84T + 41T^{2} \) |
| 43 | \( 1 + 10.5T + 43T^{2} \) |
| 47 | \( 1 - 8.95T + 47T^{2} \) |
| 53 | \( 1 + 9.97T + 53T^{2} \) |
| 59 | \( 1 - 6.38T + 59T^{2} \) |
| 61 | \( 1 - 7.23T + 61T^{2} \) |
| 67 | \( 1 - 6.02T + 67T^{2} \) |
| 71 | \( 1 - 6.71T + 71T^{2} \) |
| 73 | \( 1 - 4.61T + 73T^{2} \) |
| 79 | \( 1 + 15.4T + 79T^{2} \) |
| 83 | \( 1 + 7.21T + 83T^{2} \) |
| 89 | \( 1 - 1.94T + 89T^{2} \) |
| 97 | \( 1 - 12.7T + 97T^{2} \) |
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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
−8.325983774598618223256516981393, −7.82058833288759854832182996422, −7.21779523289452352764770312488, −6.54399871835400193535729067619, −5.04072766322631917644836815305, −4.21231642188108902953228823541, −3.24240970123879959557683598445, −2.45069611208968180708155784407, −1.03958713937779628184549352906, 0,
1.03958713937779628184549352906, 2.45069611208968180708155784407, 3.24240970123879959557683598445, 4.21231642188108902953228823541, 5.04072766322631917644836815305, 6.54399871835400193535729067619, 7.21779523289452352764770312488, 7.82058833288759854832182996422, 8.325983774598618223256516981393
