| L(s) = 1 | − 0.816·2-s − 1.53·3-s − 1.33·4-s − 5-s + 1.25·6-s + 5.03·7-s + 2.72·8-s − 0.633·9-s + 0.816·10-s − 3.03·11-s + 2.05·12-s + 4.57·13-s − 4.11·14-s + 1.53·15-s + 0.443·16-s − 1.07·17-s + 0.517·18-s + 1.33·20-s − 7.74·21-s + 2.47·22-s + 4.11·23-s − 4.18·24-s + 25-s − 3.73·26-s + 5.58·27-s − 6.71·28-s + 1.07·29-s + ⋯ |
| L(s) = 1 | − 0.577·2-s − 0.888·3-s − 0.666·4-s − 0.447·5-s + 0.512·6-s + 1.90·7-s + 0.962·8-s − 0.211·9-s + 0.258·10-s − 0.914·11-s + 0.592·12-s + 1.26·13-s − 1.09·14-s + 0.397·15-s + 0.110·16-s − 0.261·17-s + 0.121·18-s + 0.298·20-s − 1.68·21-s + 0.528·22-s + 0.857·23-s − 0.854·24-s + 0.200·25-s − 0.732·26-s + 1.07·27-s − 1.26·28-s + 0.199·29-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
| \(L(1)\) |
\(\approx\) |
\(0.7638238984\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7638238984\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.816T + 2T^{2} \) |
| 3 | \( 1 + 1.53T + 3T^{2} \) |
| 7 | \( 1 - 5.03T + 7T^{2} \) |
| 11 | \( 1 + 3.03T + 11T^{2} \) |
| 13 | \( 1 - 4.57T + 13T^{2} \) |
| 17 | \( 1 + 1.07T + 17T^{2} \) |
| 23 | \( 1 - 4.11T + 23T^{2} \) |
| 29 | \( 1 - 1.07T + 29T^{2} \) |
| 31 | \( 1 + 5.58T + 31T^{2} \) |
| 37 | \( 1 + 0.0947T + 37T^{2} \) |
| 41 | \( 1 + 10.6T + 41T^{2} \) |
| 43 | \( 1 - 5.03T + 43T^{2} \) |
| 47 | \( 1 + 12.2T + 47T^{2} \) |
| 53 | \( 1 - 4.09T + 53T^{2} \) |
| 59 | \( 1 - 1.39T + 59T^{2} \) |
| 61 | \( 1 + 5.69T + 61T^{2} \) |
| 67 | \( 1 + 5.28T + 67T^{2} \) |
| 71 | \( 1 - 5.67T + 71T^{2} \) |
| 73 | \( 1 - 9.07T + 73T^{2} \) |
| 79 | \( 1 - 5.39T + 79T^{2} \) |
| 83 | \( 1 - 1.95T + 83T^{2} \) |
| 89 | \( 1 - 2.18T + 89T^{2} \) |
| 97 | \( 1 - 2.16T + 97T^{2} \) |
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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
−8.972808825007558512513028883080, −8.441413363491454874637127008253, −7.969763324793103861646477909223, −7.11648113199396209913687111074, −5.86350402135829098367204157809, −5.00474620690255082167397730928, −4.73561164895114785653516741704, −3.50686542365179419950908498446, −1.79377811474673260537343947752, −0.71535563597378505722975222113,
0.71535563597378505722975222113, 1.79377811474673260537343947752, 3.50686542365179419950908498446, 4.73561164895114785653516741704, 5.00474620690255082167397730928, 5.86350402135829098367204157809, 7.11648113199396209913687111074, 7.969763324793103861646477909223, 8.441413363491454874637127008253, 8.972808825007558512513028883080
