| L(s) = 1 | − 1.42·2-s − 6.49·3-s − 5.95·4-s + 0.801·5-s + 9.29·6-s + 16.7·7-s + 19.9·8-s + 15.2·9-s − 1.14·10-s + 11·11-s + 38.7·12-s − 23.8·14-s − 5.20·15-s + 19.1·16-s + 43.7·17-s − 21.7·18-s + 10.5·19-s − 4.77·20-s − 108.·21-s − 15.7·22-s − 157.·23-s − 129.·24-s − 124.·25-s + 76.4·27-s − 99.5·28-s + 52.1·29-s + 7.44·30-s + ⋯ |
| L(s) = 1 | − 0.505·2-s − 1.25·3-s − 0.744·4-s + 0.0716·5-s + 0.632·6-s + 0.902·7-s + 0.881·8-s + 0.564·9-s − 0.0362·10-s + 0.301·11-s + 0.931·12-s − 0.456·14-s − 0.0896·15-s + 0.298·16-s + 0.623·17-s − 0.285·18-s + 0.126·19-s − 0.0533·20-s − 1.12·21-s − 0.152·22-s − 1.42·23-s − 1.10·24-s − 0.994·25-s + 0.544·27-s − 0.671·28-s + 0.333·29-s + 0.0453·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - 11T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 1.42T + 8T^{2} \) |
| 3 | \( 1 + 6.49T + 27T^{2} \) |
| 5 | \( 1 - 0.801T + 125T^{2} \) |
| 7 | \( 1 - 16.7T + 343T^{2} \) |
| 17 | \( 1 - 43.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 10.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 157.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 52.1T + 2.43e4T^{2} \) |
| 31 | \( 1 + 212.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 23.2T + 5.06e4T^{2} \) |
| 41 | \( 1 + 59.3T + 6.89e4T^{2} \) |
| 43 | \( 1 - 337.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 364.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 493.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 414.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 322.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 39.5T + 3.00e5T^{2} \) |
| 71 | \( 1 - 319.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 213.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 786.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 677.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 336.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.23e3T + 9.12e5T^{2} \) |
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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
−8.379536722530679768293509797922, −7.87647487188637834231148233851, −6.94657335236122045060490402613, −5.80782218634130837132994710419, −5.40010986007684116385801862075, −4.50626012376120930115841249675, −3.74659623892361802300205024817, −1.94397848103775486182354935033, −0.971154956278602896662745051765, 0,
0.971154956278602896662745051765, 1.94397848103775486182354935033, 3.74659623892361802300205024817, 4.50626012376120930115841249675, 5.40010986007684116385801862075, 5.80782218634130837132994710419, 6.94657335236122045060490402613, 7.87647487188637834231148233851, 8.379536722530679768293509797922
