| L(s) = 1 | + 2.63·2-s + 4.94·4-s − 3.15·5-s + 2.40·7-s + 7.76·8-s − 8.31·10-s + 5.19·11-s − 0.564·13-s + 6.32·14-s + 10.5·16-s + 4.49·17-s + 3.94·19-s − 15.5·20-s + 13.6·22-s − 4.30·23-s + 4.94·25-s − 1.48·26-s + 11.8·28-s − 5.83·29-s + 7.84·31-s + 12.3·32-s + 11.8·34-s − 7.57·35-s − 6.90·37-s + 10.4·38-s − 24.4·40-s − 9.60·41-s + ⋯ |
| L(s) = 1 | + 1.86·2-s + 2.47·4-s − 1.41·5-s + 0.907·7-s + 2.74·8-s − 2.62·10-s + 1.56·11-s − 0.156·13-s + 1.69·14-s + 2.64·16-s + 1.09·17-s + 0.905·19-s − 3.48·20-s + 2.91·22-s − 0.897·23-s + 0.988·25-s − 0.291·26-s + 2.24·28-s − 1.08·29-s + 1.40·31-s + 2.18·32-s + 2.03·34-s − 1.27·35-s − 1.13·37-s + 1.68·38-s − 3.87·40-s − 1.50·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6021 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6021 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.953841808\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.953841808\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 223 | \( 1 - T \) |
| good | 2 | \( 1 - 2.63T + 2T^{2} \) |
| 5 | \( 1 + 3.15T + 5T^{2} \) |
| 7 | \( 1 - 2.40T + 7T^{2} \) |
| 11 | \( 1 - 5.19T + 11T^{2} \) |
| 13 | \( 1 + 0.564T + 13T^{2} \) |
| 17 | \( 1 - 4.49T + 17T^{2} \) |
| 19 | \( 1 - 3.94T + 19T^{2} \) |
| 23 | \( 1 + 4.30T + 23T^{2} \) |
| 29 | \( 1 + 5.83T + 29T^{2} \) |
| 31 | \( 1 - 7.84T + 31T^{2} \) |
| 37 | \( 1 + 6.90T + 37T^{2} \) |
| 41 | \( 1 + 9.60T + 41T^{2} \) |
| 43 | \( 1 + 4.22T + 43T^{2} \) |
| 47 | \( 1 - 0.461T + 47T^{2} \) |
| 53 | \( 1 - 5.48T + 53T^{2} \) |
| 59 | \( 1 - 7.44T + 59T^{2} \) |
| 61 | \( 1 - 7.46T + 61T^{2} \) |
| 67 | \( 1 - 4.31T + 67T^{2} \) |
| 71 | \( 1 - 5.20T + 71T^{2} \) |
| 73 | \( 1 - 16.4T + 73T^{2} \) |
| 79 | \( 1 - 8.49T + 79T^{2} \) |
| 83 | \( 1 + 0.852T + 83T^{2} \) |
| 89 | \( 1 + 9.86T + 89T^{2} \) |
| 97 | \( 1 - 10.8T + 97T^{2} \) |
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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
−7.993117225700566714356792806876, −7.02221809648328406492542155612, −6.73394258575409805283794648698, −5.63596279675989389661939597171, −5.07634093186471392348602373406, −4.35860683381551478739788677461, −3.54861421719306647702864722885, −3.53383650083895469956919345236, −2.09077895982879711731196945972, −1.14854569768155668941883314366,
1.14854569768155668941883314366, 2.09077895982879711731196945972, 3.53383650083895469956919345236, 3.54861421719306647702864722885, 4.35860683381551478739788677461, 5.07634093186471392348602373406, 5.63596279675989389661939597171, 6.73394258575409805283794648698, 7.02221809648328406492542155612, 7.993117225700566714356792806876
