| L(s) = 1 | − 1.19·2-s + 3.04·3-s − 0.581·4-s + 5-s − 3.62·6-s − 0.609·7-s + 3.07·8-s + 6.28·9-s − 1.19·10-s + 4.48·11-s − 1.77·12-s + 4.43·13-s + 0.725·14-s + 3.04·15-s − 2.49·16-s + 2.90·17-s − 7.48·18-s − 0.581·20-s − 1.85·21-s − 5.34·22-s − 2.84·23-s + 9.36·24-s + 25-s − 5.28·26-s + 10.0·27-s + 0.354·28-s + 1.11·29-s + ⋯ |
| L(s) = 1 | − 0.842·2-s + 1.75·3-s − 0.290·4-s + 0.447·5-s − 1.48·6-s − 0.230·7-s + 1.08·8-s + 2.09·9-s − 0.376·10-s + 1.35·11-s − 0.511·12-s + 1.23·13-s + 0.193·14-s + 0.786·15-s − 0.624·16-s + 0.704·17-s − 1.76·18-s − 0.130·20-s − 0.405·21-s − 1.13·22-s − 0.593·23-s + 1.91·24-s + 0.200·25-s − 1.03·26-s + 1.92·27-s + 0.0669·28-s + 0.207·29-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
| \(L(1)\) |
\(\approx\) |
\(2.358493137\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.358493137\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.19T + 2T^{2} \) |
| 3 | \( 1 - 3.04T + 3T^{2} \) |
| 7 | \( 1 + 0.609T + 7T^{2} \) |
| 11 | \( 1 - 4.48T + 11T^{2} \) |
| 13 | \( 1 - 4.43T + 13T^{2} \) |
| 17 | \( 1 - 2.90T + 17T^{2} \) |
| 23 | \( 1 + 2.84T + 23T^{2} \) |
| 29 | \( 1 - 1.11T + 29T^{2} \) |
| 31 | \( 1 + 6.22T + 31T^{2} \) |
| 37 | \( 1 + 3.77T + 37T^{2} \) |
| 41 | \( 1 + 8.30T + 41T^{2} \) |
| 43 | \( 1 + 9.98T + 43T^{2} \) |
| 47 | \( 1 + 5.88T + 47T^{2} \) |
| 53 | \( 1 - 8.44T + 53T^{2} \) |
| 59 | \( 1 - 10.2T + 59T^{2} \) |
| 61 | \( 1 + 4.98T + 61T^{2} \) |
| 67 | \( 1 - 8.47T + 67T^{2} \) |
| 71 | \( 1 - 11.6T + 71T^{2} \) |
| 73 | \( 1 - 3.72T + 73T^{2} \) |
| 79 | \( 1 - 9.03T + 79T^{2} \) |
| 83 | \( 1 + 2.12T + 83T^{2} \) |
| 89 | \( 1 - 7.93T + 89T^{2} \) |
| 97 | \( 1 + 9.67T + 97T^{2} \) |
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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
−9.186398665007881909931929701588, −8.485165806935015430826822184027, −8.222270959542748869425981394801, −7.14672383282260545714046161219, −6.44464758554663210699469466173, −5.06582444282989457149768264812, −3.69029952371709733739284118189, −3.60886012544964603176212143468, −1.96274365843339118862116095577, −1.28786942053371100008424776250,
1.28786942053371100008424776250, 1.96274365843339118862116095577, 3.60886012544964603176212143468, 3.69029952371709733739284118189, 5.06582444282989457149768264812, 6.44464758554663210699469466173, 7.14672383282260545714046161219, 8.222270959542748869425981394801, 8.485165806935015430826822184027, 9.186398665007881909931929701588
