| L(s) = 1 | − 8.68·2-s − 13.3·3-s + 43.4·4-s − 25·5-s + 116.·6-s − 102.·7-s − 99.1·8-s − 63.7·9-s + 217.·10-s − 121·11-s − 581.·12-s + 752.·13-s + 888.·14-s + 334.·15-s − 528.·16-s + 1.29e3·17-s + 553.·18-s + 361·19-s − 1.08e3·20-s + 1.37e3·21-s + 1.05e3·22-s − 194.·23-s + 1.32e3·24-s + 625·25-s − 6.53e3·26-s + 4.10e3·27-s − 4.44e3·28-s + ⋯ |
| L(s) = 1 | − 1.53·2-s − 0.858·3-s + 1.35·4-s − 0.447·5-s + 1.31·6-s − 0.789·7-s − 0.547·8-s − 0.262·9-s + 0.686·10-s − 0.301·11-s − 1.16·12-s + 1.23·13-s + 1.21·14-s + 0.384·15-s − 0.516·16-s + 1.08·17-s + 0.402·18-s + 0.229·19-s − 0.606·20-s + 0.677·21-s + 0.462·22-s − 0.0767·23-s + 0.470·24-s + 0.200·25-s − 1.89·26-s + 1.08·27-s − 1.07·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.5564775982\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5564775982\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 25T \) |
| 11 | \( 1 + 121T \) |
| 19 | \( 1 - 361T \) |
| good | 2 | \( 1 + 8.68T + 32T^{2} \) |
| 3 | \( 1 + 13.3T + 243T^{2} \) |
| 7 | \( 1 + 102.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 752.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.29e3T + 1.41e6T^{2} \) |
| 23 | \( 1 + 194.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.00e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 4.19e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 6.02e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.95e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 3.49e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.36e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.36e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.00e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.67e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.49e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.23e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.83e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 9.50e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 4.66e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.17e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 7.01e4T + 8.58e9T^{2} \) |
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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
−9.312594201241807389242858984023, −8.253376529635020705342352816723, −7.87546714736636738573777642740, −6.67074544011243655369172118381, −6.17087793476916623227107142934, −5.10510214741010880379758701592, −3.73359287074784281278856513228, −2.65618987134653710133856355236, −1.09551370921008192020726240332, −0.53618453772101905326399509333,
0.53618453772101905326399509333, 1.09551370921008192020726240332, 2.65618987134653710133856355236, 3.73359287074784281278856513228, 5.10510214741010880379758701592, 6.17087793476916623227107142934, 6.67074544011243655369172118381, 7.87546714736636738573777642740, 8.253376529635020705342352816723, 9.312594201241807389242858984023
