L(s) = 1 | + 3.27·3-s − 1.49·5-s + 7.70·9-s + 5.00·11-s + 2.30·13-s − 4.90·15-s + 5.61·17-s − 19-s + 6.90·23-s − 2.75·25-s + 15.4·27-s + 5.82·29-s − 7.98·31-s + 16.3·33-s − 3.77·37-s + 7.54·39-s + 8.85·41-s − 12.4·43-s − 11.5·45-s + 8.73·47-s + 18.3·51-s − 11.2·53-s − 7.49·55-s − 3.27·57-s − 7.65·59-s − 4.11·61-s − 3.45·65-s + ⋯ |
L(s) = 1 | + 1.88·3-s − 0.670·5-s + 2.56·9-s + 1.50·11-s + 0.639·13-s − 1.26·15-s + 1.36·17-s − 0.229·19-s + 1.44·23-s − 0.550·25-s + 2.96·27-s + 1.08·29-s − 1.43·31-s + 2.84·33-s − 0.620·37-s + 1.20·39-s + 1.38·41-s − 1.89·43-s − 1.72·45-s + 1.27·47-s + 2.57·51-s − 1.54·53-s − 1.01·55-s − 0.433·57-s − 0.996·59-s − 0.526·61-s − 0.428·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.894999908\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.894999908\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 - 3.27T + 3T^{2} \) |
| 5 | \( 1 + 1.49T + 5T^{2} \) |
| 11 | \( 1 - 5.00T + 11T^{2} \) |
| 13 | \( 1 - 2.30T + 13T^{2} \) |
| 17 | \( 1 - 5.61T + 17T^{2} \) |
| 23 | \( 1 - 6.90T + 23T^{2} \) |
| 29 | \( 1 - 5.82T + 29T^{2} \) |
| 31 | \( 1 + 7.98T + 31T^{2} \) |
| 37 | \( 1 + 3.77T + 37T^{2} \) |
| 41 | \( 1 - 8.85T + 41T^{2} \) |
| 43 | \( 1 + 12.4T + 43T^{2} \) |
| 47 | \( 1 - 8.73T + 47T^{2} \) |
| 53 | \( 1 + 11.2T + 53T^{2} \) |
| 59 | \( 1 + 7.65T + 59T^{2} \) |
| 61 | \( 1 + 4.11T + 61T^{2} \) |
| 67 | \( 1 + 5.67T + 67T^{2} \) |
| 71 | \( 1 + 9.70T + 71T^{2} \) |
| 73 | \( 1 + 2.17T + 73T^{2} \) |
| 79 | \( 1 + 4.54T + 79T^{2} \) |
| 83 | \( 1 + 8.37T + 83T^{2} \) |
| 89 | \( 1 - 17.3T + 89T^{2} \) |
| 97 | \( 1 + 1.01T + 97T^{2} \) |
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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
−7.936195077619121883588348471886, −7.39356848893286966096205571038, −6.83253918891738616745871336667, −5.93732681108303124191513468385, −4.69394776921182894222353484913, −4.05451147215367865703123044129, −3.37271151253112989802786496161, −3.06608443414889498815012141248, −1.74823072364957601254240983531, −1.15754866005462040492348337075,
1.15754866005462040492348337075, 1.74823072364957601254240983531, 3.06608443414889498815012141248, 3.37271151253112989802786496161, 4.05451147215367865703123044129, 4.69394776921182894222353484913, 5.93732681108303124191513468385, 6.83253918891738616745871336667, 7.39356848893286966096205571038, 7.936195077619121883588348471886