| L(s) = 1 | + 2.50·2-s − 1.22·3-s + 4.28·4-s − 5-s − 3.06·6-s − 0.221·7-s + 5.72·8-s − 1.50·9-s − 2.50·10-s − 0.778·11-s − 5.23·12-s + 5·13-s − 0.556·14-s + 1.22·15-s + 5.79·16-s + 7.07·17-s − 3.77·18-s − 4.28·20-s + 0.271·21-s − 1.95·22-s + 8.07·23-s − 6.99·24-s + 25-s + 12.5·26-s + 5.50·27-s − 0.950·28-s − 0.221·29-s + ⋯ |
| L(s) = 1 | + 1.77·2-s − 0.705·3-s + 2.14·4-s − 0.447·5-s − 1.25·6-s − 0.0838·7-s + 2.02·8-s − 0.502·9-s − 0.792·10-s − 0.234·11-s − 1.51·12-s + 1.38·13-s − 0.148·14-s + 0.315·15-s + 1.44·16-s + 1.71·17-s − 0.890·18-s − 0.958·20-s + 0.0591·21-s − 0.415·22-s + 1.68·23-s − 1.42·24-s + 0.200·25-s + 2.45·26-s + 1.05·27-s − 0.179·28-s − 0.0412·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.922163657\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.922163657\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - 2.50T + 2T^{2} \) |
| 3 | \( 1 + 1.22T + 3T^{2} \) |
| 7 | \( 1 + 0.221T + 7T^{2} \) |
| 11 | \( 1 + 0.778T + 11T^{2} \) |
| 13 | \( 1 - 5T + 13T^{2} \) |
| 17 | \( 1 - 7.07T + 17T^{2} \) |
| 23 | \( 1 - 8.07T + 23T^{2} \) |
| 29 | \( 1 + 0.221T + 29T^{2} \) |
| 31 | \( 1 + 2.50T + 31T^{2} \) |
| 37 | \( 1 - 1.90T + 37T^{2} \) |
| 41 | \( 1 - 7.23T + 41T^{2} \) |
| 43 | \( 1 - 7.29T + 43T^{2} \) |
| 47 | \( 1 + 2.79T + 47T^{2} \) |
| 53 | \( 1 + 4.38T + 53T^{2} \) |
| 59 | \( 1 - 2.79T + 59T^{2} \) |
| 61 | \( 1 + 12.5T + 61T^{2} \) |
| 67 | \( 1 - 10.5T + 67T^{2} \) |
| 71 | \( 1 - 9.84T + 71T^{2} \) |
| 73 | \( 1 + 14.0T + 73T^{2} \) |
| 79 | \( 1 + 1.58T + 79T^{2} \) |
| 83 | \( 1 + 9.52T + 83T^{2} \) |
| 89 | \( 1 + 3.14T + 89T^{2} \) |
| 97 | \( 1 - 6.36T + 97T^{2} \) |
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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
−9.304949834538538340979393740436, −8.222989055078118401426871969071, −7.36947446937082273637432943860, −6.48424287429884955539389438704, −5.78318229225283275498511470231, −5.31007713684764805376829016097, −4.40019781541403609919521142526, −3.42506482079454655969003585468, −2.89225026118969653358759324495, −1.16585413015734357875497579948,
1.16585413015734357875497579948, 2.89225026118969653358759324495, 3.42506482079454655969003585468, 4.40019781541403609919521142526, 5.31007713684764805376829016097, 5.78318229225283275498511470231, 6.48424287429884955539389438704, 7.36947446937082273637432943860, 8.222989055078118401426871969071, 9.304949834538538340979393740436
