| L(s) = 1 | − 4.48·2-s + 3·3-s + 12.1·4-s + 5.33·5-s − 13.4·6-s + 16.1·7-s − 18.6·8-s + 9·9-s − 23.9·10-s − 11·11-s + 36.4·12-s − 61.5·13-s − 72.3·14-s + 15.9·15-s − 13.4·16-s − 34.0·17-s − 40.4·18-s − 1.73·19-s + 64.7·20-s + 48.3·21-s + 49.3·22-s + 65.5·23-s − 55.9·24-s − 96.5·25-s + 276.·26-s + 27·27-s + 195.·28-s + ⋯ |
| L(s) = 1 | − 1.58·2-s + 0.577·3-s + 1.51·4-s + 0.476·5-s − 0.916·6-s + 0.869·7-s − 0.824·8-s + 0.333·9-s − 0.756·10-s − 0.301·11-s + 0.877·12-s − 1.31·13-s − 1.38·14-s + 0.275·15-s − 0.210·16-s − 0.485·17-s − 0.529·18-s − 0.0209·19-s + 0.724·20-s + 0.502·21-s + 0.478·22-s + 0.594·23-s − 0.476·24-s − 0.772·25-s + 2.08·26-s + 0.192·27-s + 1.32·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - 3T \) |
| 11 | \( 1 + 11T \) |
| 61 | \( 1 + 61T \) |
| good | 2 | \( 1 + 4.48T + 8T^{2} \) |
| 5 | \( 1 - 5.33T + 125T^{2} \) |
| 7 | \( 1 - 16.1T + 343T^{2} \) |
| 13 | \( 1 + 61.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 34.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 1.73T + 6.85e3T^{2} \) |
| 23 | \( 1 - 65.5T + 1.21e4T^{2} \) |
| 29 | \( 1 - 198.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 115.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 425.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 394.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 291.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 398.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 316.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 609.T + 2.05e5T^{2} \) |
| 67 | \( 1 - 403.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 849.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 993.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 548.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 876.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 544.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 986.T + 9.12e5T^{2} \) |
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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
−8.372300333017828069899847955824, −7.894592921528181360272677832645, −7.23053343234667455561028573191, −6.42181505309951840744795649141, −5.15161555877377026232342245445, −4.39376644541578016954164265140, −2.78154447193786368043425984181, −2.12298626635548261040891528047, −1.25523671109123638324230781723, 0,
1.25523671109123638324230781723, 2.12298626635548261040891528047, 2.78154447193786368043425984181, 4.39376644541578016954164265140, 5.15161555877377026232342245445, 6.42181505309951840744795649141, 7.23053343234667455561028573191, 7.894592921528181360272677832645, 8.372300333017828069899847955824
