| L(s) = 1 | − 2.44·2-s + 0.992·3-s + 3.99·4-s − 2.42·6-s − 0.744·7-s − 4.88·8-s − 2.01·9-s − 2.60·11-s + 3.96·12-s + 4.18·13-s + 1.82·14-s + 3.96·16-s + 0.484·17-s + 4.93·18-s + 3.30·19-s − 0.738·21-s + 6.37·22-s − 1.44·23-s − 4.84·24-s − 10.2·26-s − 4.97·27-s − 2.97·28-s − 0.129·29-s + 4.43·31-s + 0.0531·32-s − 2.58·33-s − 1.18·34-s + ⋯ |
| L(s) = 1 | − 1.73·2-s + 0.572·3-s + 1.99·4-s − 0.991·6-s − 0.281·7-s − 1.72·8-s − 0.671·9-s − 0.785·11-s + 1.14·12-s + 1.16·13-s + 0.487·14-s + 0.991·16-s + 0.117·17-s + 1.16·18-s + 0.757·19-s − 0.161·21-s + 1.35·22-s − 0.300·23-s − 0.989·24-s − 2.00·26-s − 0.957·27-s − 0.562·28-s − 0.0240·29-s + 0.796·31-s + 0.00939·32-s − 0.449·33-s − 0.203·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 241 | \( 1 - T \) |
| good | 2 | \( 1 + 2.44T + 2T^{2} \) |
| 3 | \( 1 - 0.992T + 3T^{2} \) |
| 7 | \( 1 + 0.744T + 7T^{2} \) |
| 11 | \( 1 + 2.60T + 11T^{2} \) |
| 13 | \( 1 - 4.18T + 13T^{2} \) |
| 17 | \( 1 - 0.484T + 17T^{2} \) |
| 19 | \( 1 - 3.30T + 19T^{2} \) |
| 23 | \( 1 + 1.44T + 23T^{2} \) |
| 29 | \( 1 + 0.129T + 29T^{2} \) |
| 31 | \( 1 - 4.43T + 31T^{2} \) |
| 37 | \( 1 - 4.67T + 37T^{2} \) |
| 41 | \( 1 + 9.21T + 41T^{2} \) |
| 43 | \( 1 + 7.65T + 43T^{2} \) |
| 47 | \( 1 + 2.28T + 47T^{2} \) |
| 53 | \( 1 - 12.5T + 53T^{2} \) |
| 59 | \( 1 + 7.30T + 59T^{2} \) |
| 61 | \( 1 + 5.28T + 61T^{2} \) |
| 67 | \( 1 - 5.69T + 67T^{2} \) |
| 71 | \( 1 - 2.08T + 71T^{2} \) |
| 73 | \( 1 - 15.5T + 73T^{2} \) |
| 79 | \( 1 + 10.0T + 79T^{2} \) |
| 83 | \( 1 + 2.71T + 83T^{2} \) |
| 89 | \( 1 + 9.86T + 89T^{2} \) |
| 97 | \( 1 - 6.26T + 97T^{2} \) |
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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
−8.065750305926808301588600033705, −7.36348198971824695106679518309, −6.52403235325140772432538948981, −5.92386364748736568058089069036, −4.97426110716583212339841897158, −3.59172240729525256023301357905, −2.95993975249728702138874117596, −2.13331510404211575818588154165, −1.14632750826167583598798646459, 0,
1.14632750826167583598798646459, 2.13331510404211575818588154165, 2.95993975249728702138874117596, 3.59172240729525256023301357905, 4.97426110716583212339841897158, 5.92386364748736568058089069036, 6.52403235325140772432538948981, 7.36348198971824695106679518309, 8.065750305926808301588600033705
