| L(s) = 1 | − 2.70·2-s − 3.35·3-s + 5.30·4-s + 9.07·6-s + 3.43·7-s − 8.94·8-s + 8.25·9-s + 4.01·11-s − 17.8·12-s + 0.273·13-s − 9.27·14-s + 13.5·16-s + 6.77·17-s − 22.3·18-s − 5.41·19-s − 11.5·21-s − 10.8·22-s + 2.33·23-s + 30.0·24-s − 0.738·26-s − 17.6·27-s + 18.2·28-s + 3.85·29-s + 0.673·31-s − 18.7·32-s − 13.4·33-s − 18.3·34-s + ⋯ |
| L(s) = 1 | − 1.91·2-s − 1.93·3-s + 2.65·4-s + 3.70·6-s + 1.29·7-s − 3.16·8-s + 2.75·9-s + 1.21·11-s − 5.14·12-s + 0.0757·13-s − 2.47·14-s + 3.38·16-s + 1.64·17-s − 5.26·18-s − 1.24·19-s − 2.51·21-s − 2.31·22-s + 0.487·23-s + 6.12·24-s − 0.144·26-s − 3.39·27-s + 3.44·28-s + 0.714·29-s + 0.120·31-s − 3.31·32-s − 2.34·33-s − 3.14·34-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
| \(L(1)\) |
\(\approx\) |
\(0.5885438014\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5885438014\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.70T + 2T^{2} \) |
| 3 | \( 1 + 3.35T + 3T^{2} \) |
| 7 | \( 1 - 3.43T + 7T^{2} \) |
| 11 | \( 1 - 4.01T + 11T^{2} \) |
| 13 | \( 1 - 0.273T + 13T^{2} \) |
| 17 | \( 1 - 6.77T + 17T^{2} \) |
| 19 | \( 1 + 5.41T + 19T^{2} \) |
| 23 | \( 1 - 2.33T + 23T^{2} \) |
| 29 | \( 1 - 3.85T + 29T^{2} \) |
| 31 | \( 1 - 0.673T + 31T^{2} \) |
| 37 | \( 1 + 4.50T + 37T^{2} \) |
| 41 | \( 1 + 3.57T + 41T^{2} \) |
| 43 | \( 1 - 0.816T + 43T^{2} \) |
| 47 | \( 1 - 1.60T + 47T^{2} \) |
| 53 | \( 1 + 4.44T + 53T^{2} \) |
| 59 | \( 1 - 0.529T + 59T^{2} \) |
| 61 | \( 1 + 6.64T + 61T^{2} \) |
| 67 | \( 1 - 7.41T + 67T^{2} \) |
| 71 | \( 1 - 14.8T + 71T^{2} \) |
| 73 | \( 1 - 3.73T + 73T^{2} \) |
| 79 | \( 1 - 3.17T + 79T^{2} \) |
| 83 | \( 1 - 13.2T + 83T^{2} \) |
| 89 | \( 1 - 1.74T + 89T^{2} \) |
| 97 | \( 1 + 1.39T + 97T^{2} \) |
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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
−8.026188178969191539496875858471, −7.45400989293941258772644637813, −6.63602489337111517547055475442, −6.33794052259619826838463027252, −5.45506238210951412795180838371, −4.74813327141181759038207275238, −3.64718927938424054433172742653, −1.96401262553282243478887612945, −1.31361278580554759175238487203, −0.70311477640708602998234196845,
0.70311477640708602998234196845, 1.31361278580554759175238487203, 1.96401262553282243478887612945, 3.64718927938424054433172742653, 4.74813327141181759038207275238, 5.45506238210951412795180838371, 6.33794052259619826838463027252, 6.63602489337111517547055475442, 7.45400989293941258772644637813, 8.026188178969191539496875858471
