| L(s) = 1 | − 9.64·3-s + 5·5-s − 34.7·7-s + 65.9·9-s − 12.5·11-s − 25.0·13-s − 48.2·15-s − 59.0·17-s − 61.8·19-s + 334.·21-s + 21.8·23-s + 25·25-s − 375.·27-s + 29·29-s − 319.·31-s + 121.·33-s − 173.·35-s − 370.·37-s + 241.·39-s − 490.·41-s + 213.·43-s + 329.·45-s − 408.·47-s + 862.·49-s + 568.·51-s + 67.8·53-s − 62.8·55-s + ⋯ |
| L(s) = 1 | − 1.85·3-s + 0.447·5-s − 1.87·7-s + 2.44·9-s − 0.344·11-s − 0.534·13-s − 0.829·15-s − 0.841·17-s − 0.746·19-s + 3.47·21-s + 0.198·23-s + 0.200·25-s − 2.67·27-s + 0.185·29-s − 1.85·31-s + 0.638·33-s − 0.838·35-s − 1.64·37-s + 0.991·39-s − 1.86·41-s + 0.758·43-s + 1.09·45-s − 1.26·47-s + 2.51·49-s + 1.56·51-s + 0.175·53-s − 0.154·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.003741953558\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.003741953558\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| 29 | \( 1 - 29T \) |
| good | 3 | \( 1 + 9.64T + 27T^{2} \) |
| 7 | \( 1 + 34.7T + 343T^{2} \) |
| 11 | \( 1 + 12.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 25.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 59.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 61.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 21.8T + 1.21e4T^{2} \) |
| 31 | \( 1 + 319.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 370.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 490.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 213.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 408.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 67.8T + 1.48e5T^{2} \) |
| 59 | \( 1 + 200.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 559.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 936.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 577.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.04e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 236.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.21e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.12e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 498.T + 9.12e5T^{2} \) |
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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
−9.670145429013768004668967709556, −8.900177438666038343083109806465, −7.17873629831697476223657439060, −6.76940155117212866247407078294, −6.05140371354414034524795934109, −5.38427966531705169744734702417, −4.44376163196853705610256767699, −3.26094953765759801370366883538, −1.79012664824546350147591471344, −0.03322278175610676689390214789,
0.03322278175610676689390214789, 1.79012664824546350147591471344, 3.26094953765759801370366883538, 4.44376163196853705610256767699, 5.38427966531705169744734702417, 6.05140371354414034524795934109, 6.76940155117212866247407078294, 7.17873629831697476223657439060, 8.900177438666038343083109806465, 9.670145429013768004668967709556
