| L(s) = 1 | − 0.917·2-s + 3·3-s − 7.15·4-s + 15.4·5-s − 2.75·6-s + 20.5·7-s + 13.9·8-s + 9·9-s − 14.1·10-s + 65.8·11-s − 21.4·12-s − 18.8·14-s + 46.4·15-s + 44.5·16-s + 44.2·17-s − 8.25·18-s − 147.·19-s − 110.·20-s + 61.7·21-s − 60.4·22-s + 53.1·23-s + 41.7·24-s + 114.·25-s + 27·27-s − 147.·28-s − 38.6·29-s − 42.5·30-s + ⋯ |
| L(s) = 1 | − 0.324·2-s + 0.577·3-s − 0.894·4-s + 1.38·5-s − 0.187·6-s + 1.11·7-s + 0.614·8-s + 0.333·9-s − 0.448·10-s + 1.80·11-s − 0.516·12-s − 0.360·14-s + 0.798·15-s + 0.695·16-s + 0.631·17-s − 0.108·18-s − 1.77·19-s − 1.23·20-s + 0.641·21-s − 0.585·22-s + 0.481·23-s + 0.354·24-s + 0.914·25-s + 0.192·27-s − 0.994·28-s − 0.247·29-s − 0.259·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.770088580\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.770088580\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 0.917T + 8T^{2} \) |
| 5 | \( 1 - 15.4T + 125T^{2} \) |
| 7 | \( 1 - 20.5T + 343T^{2} \) |
| 11 | \( 1 - 65.8T + 1.33e3T^{2} \) |
| 17 | \( 1 - 44.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 147.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 53.1T + 1.21e4T^{2} \) |
| 29 | \( 1 + 38.6T + 2.43e4T^{2} \) |
| 31 | \( 1 + 88.3T + 2.97e4T^{2} \) |
| 37 | \( 1 + 78.9T + 5.06e4T^{2} \) |
| 41 | \( 1 - 354.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 407.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 67.9T + 1.03e5T^{2} \) |
| 53 | \( 1 - 226.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 142.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 266.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 411.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 91.5T + 3.57e5T^{2} \) |
| 73 | \( 1 + 63.1T + 3.89e5T^{2} \) |
| 79 | \( 1 + 287.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 373.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 119.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 554.T + 9.12e5T^{2} \) |
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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
−10.22821157651500748587806054062, −9.416565162703171947596371274744, −8.865336173112898250506508209493, −8.161334390673380098776795153094, −6.87532962732145192597307939748, −5.80453800889282147016586977239, −4.72105609149777327212502036212, −3.80737142694994287557392764061, −1.99899021887524170589916739665, −1.23330040329395130942763748572,
1.23330040329395130942763748572, 1.99899021887524170589916739665, 3.80737142694994287557392764061, 4.72105609149777327212502036212, 5.80453800889282147016586977239, 6.87532962732145192597307939748, 8.161334390673380098776795153094, 8.865336173112898250506508209493, 9.416565162703171947596371274744, 10.22821157651500748587806054062
