| L(s) = 1 | − 2-s + 2.68·3-s + 4-s − 0.124·5-s − 2.68·6-s + 4.08·7-s − 8-s + 4.22·9-s + 0.124·10-s − 2.67·11-s + 2.68·12-s + 4.07·13-s − 4.08·14-s − 0.333·15-s + 16-s − 3.86·17-s − 4.22·18-s + 3.98·19-s − 0.124·20-s + 10.9·21-s + 2.67·22-s − 3.03·23-s − 2.68·24-s − 4.98·25-s − 4.07·26-s + 3.27·27-s + 4.08·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.55·3-s + 0.5·4-s − 0.0554·5-s − 1.09·6-s + 1.54·7-s − 0.353·8-s + 1.40·9-s + 0.0392·10-s − 0.805·11-s + 0.775·12-s + 1.13·13-s − 1.09·14-s − 0.0860·15-s + 0.250·16-s − 0.937·17-s − 0.994·18-s + 0.914·19-s − 0.0277·20-s + 2.39·21-s + 0.569·22-s − 0.633·23-s − 0.548·24-s − 0.996·25-s − 0.799·26-s + 0.630·27-s + 0.772·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.183468202\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.183468202\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| 97 | \( 1 + T \) |
| good | 3 | \( 1 - 2.68T + 3T^{2} \) |
| 5 | \( 1 + 0.124T + 5T^{2} \) |
| 7 | \( 1 - 4.08T + 7T^{2} \) |
| 11 | \( 1 + 2.67T + 11T^{2} \) |
| 13 | \( 1 - 4.07T + 13T^{2} \) |
| 17 | \( 1 + 3.86T + 17T^{2} \) |
| 19 | \( 1 - 3.98T + 19T^{2} \) |
| 23 | \( 1 + 3.03T + 23T^{2} \) |
| 29 | \( 1 + 5.69T + 29T^{2} \) |
| 37 | \( 1 - 3.45T + 37T^{2} \) |
| 41 | \( 1 + 8.86T + 41T^{2} \) |
| 43 | \( 1 - 10.6T + 43T^{2} \) |
| 47 | \( 1 - 11.6T + 47T^{2} \) |
| 53 | \( 1 - 10.8T + 53T^{2} \) |
| 59 | \( 1 - 12.6T + 59T^{2} \) |
| 61 | \( 1 - 9.53T + 61T^{2} \) |
| 67 | \( 1 - 2.88T + 67T^{2} \) |
| 71 | \( 1 - 2.82T + 71T^{2} \) |
| 73 | \( 1 + 3.70T + 73T^{2} \) |
| 79 | \( 1 - 4.63T + 79T^{2} \) |
| 83 | \( 1 - 10.7T + 83T^{2} \) |
| 89 | \( 1 + 11.3T + 89T^{2} \) |
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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
−8.220006868454515260245743711249, −7.59059329236996727040793838806, −7.23788932109555253901947505249, −5.95313038074845992190289874216, −5.23534677902858786640534103325, −4.11364240057625821944503177592, −3.63194434583118451703302945930, −2.33983739017543413664243887536, −2.11213060496450744140593114952, −1.00948780864068293068918267387,
1.00948780864068293068918267387, 2.11213060496450744140593114952, 2.33983739017543413664243887536, 3.63194434583118451703302945930, 4.11364240057625821944503177592, 5.23534677902858786640534103325, 5.95313038074845992190289874216, 7.23788932109555253901947505249, 7.59059329236996727040793838806, 8.220006868454515260245743711249
