| L(s) = 1 | + 1.23·2-s + 3.02·3-s − 0.477·4-s + 3.72·6-s − 0.346·7-s − 3.05·8-s + 6.12·9-s + 3.53·11-s − 1.44·12-s + 0.754·13-s − 0.427·14-s − 2.81·16-s − 6.52·17-s + 7.56·18-s + 2.80·19-s − 1.04·21-s + 4.36·22-s + 5.53·23-s − 9.23·24-s + 0.931·26-s + 9.45·27-s + 0.165·28-s + 10.6·29-s − 7.44·31-s + 2.63·32-s + 10.6·33-s − 8.05·34-s + ⋯ |
| L(s) = 1 | + 0.872·2-s + 1.74·3-s − 0.238·4-s + 1.52·6-s − 0.131·7-s − 1.08·8-s + 2.04·9-s + 1.06·11-s − 0.416·12-s + 0.209·13-s − 0.114·14-s − 0.704·16-s − 1.58·17-s + 1.78·18-s + 0.644·19-s − 0.228·21-s + 0.930·22-s + 1.15·23-s − 1.88·24-s + 0.182·26-s + 1.81·27-s + 0.0312·28-s + 1.96·29-s − 1.33·31-s + 0.466·32-s + 1.85·33-s − 1.38·34-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
| \(L(1)\) |
\(\approx\) |
\(5.628548060\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.628548060\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.23T + 2T^{2} \) |
| 3 | \( 1 - 3.02T + 3T^{2} \) |
| 7 | \( 1 + 0.346T + 7T^{2} \) |
| 11 | \( 1 - 3.53T + 11T^{2} \) |
| 13 | \( 1 - 0.754T + 13T^{2} \) |
| 17 | \( 1 + 6.52T + 17T^{2} \) |
| 19 | \( 1 - 2.80T + 19T^{2} \) |
| 23 | \( 1 - 5.53T + 23T^{2} \) |
| 29 | \( 1 - 10.6T + 29T^{2} \) |
| 31 | \( 1 + 7.44T + 31T^{2} \) |
| 37 | \( 1 - 8.79T + 37T^{2} \) |
| 41 | \( 1 - 9.37T + 41T^{2} \) |
| 43 | \( 1 - 2.18T + 43T^{2} \) |
| 47 | \( 1 + 0.847T + 47T^{2} \) |
| 53 | \( 1 - 2.80T + 53T^{2} \) |
| 59 | \( 1 + 8.40T + 59T^{2} \) |
| 61 | \( 1 + 8.32T + 61T^{2} \) |
| 67 | \( 1 - 2.43T + 67T^{2} \) |
| 71 | \( 1 + 4.81T + 71T^{2} \) |
| 73 | \( 1 - 8.19T + 73T^{2} \) |
| 79 | \( 1 + 0.918T + 79T^{2} \) |
| 83 | \( 1 + 0.583T + 83T^{2} \) |
| 89 | \( 1 - 13.9T + 89T^{2} \) |
| 97 | \( 1 + 2.05T + 97T^{2} \) |
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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
−8.226327029831743087348445926284, −7.38082302620859059956538648113, −6.64927336282601554359524498018, −6.02579142239330451867195572333, −4.70751522943934343801081694787, −4.40414224566959742192732112396, −3.58028311944446149246570187285, −2.99823545728954735357991855528, −2.28041032141798154047398261887, −1.06536032174000062635669977735,
1.06536032174000062635669977735, 2.28041032141798154047398261887, 2.99823545728954735357991855528, 3.58028311944446149246570187285, 4.40414224566959742192732112396, 4.70751522943934343801081694787, 6.02579142239330451867195572333, 6.64927336282601554359524498018, 7.38082302620859059956538648113, 8.226327029831743087348445926284
