| L(s) = 1 | − 0.875·2-s + 2.02·3-s − 1.23·4-s − 3.23·5-s − 1.76·6-s − 2.40·7-s + 2.83·8-s + 1.08·9-s + 2.83·10-s − 6.21·11-s − 2.49·12-s − 0.604·13-s + 2.10·14-s − 6.54·15-s − 0.0101·16-s + 3.61·17-s − 0.946·18-s − 2.17·19-s + 3.99·20-s − 4.84·21-s + 5.44·22-s + 3.45·23-s + 5.71·24-s + 5.49·25-s + 0.529·26-s − 3.87·27-s + 2.96·28-s + ⋯ |
| L(s) = 1 | − 0.618·2-s + 1.16·3-s − 0.616·4-s − 1.44·5-s − 0.721·6-s − 0.907·7-s + 1.00·8-s + 0.360·9-s + 0.896·10-s − 1.87·11-s − 0.719·12-s − 0.167·13-s + 0.561·14-s − 1.68·15-s − 0.00254·16-s + 0.875·17-s − 0.223·18-s − 0.499·19-s + 0.893·20-s − 1.05·21-s + 1.16·22-s + 0.719·23-s + 1.16·24-s + 1.09·25-s + 0.103·26-s − 0.745·27-s + 0.559·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9409 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9409 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.003752065179\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.003752065179\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 97 | \( 1 \) |
| good | 2 | \( 1 + 0.875T + 2T^{2} \) |
| 3 | \( 1 - 2.02T + 3T^{2} \) |
| 5 | \( 1 + 3.23T + 5T^{2} \) |
| 7 | \( 1 + 2.40T + 7T^{2} \) |
| 11 | \( 1 + 6.21T + 11T^{2} \) |
| 13 | \( 1 + 0.604T + 13T^{2} \) |
| 17 | \( 1 - 3.61T + 17T^{2} \) |
| 19 | \( 1 + 2.17T + 19T^{2} \) |
| 23 | \( 1 - 3.45T + 23T^{2} \) |
| 29 | \( 1 + 3.95T + 29T^{2} \) |
| 31 | \( 1 + 6.61T + 31T^{2} \) |
| 37 | \( 1 + 4.61T + 37T^{2} \) |
| 41 | \( 1 - 3.85T + 41T^{2} \) |
| 43 | \( 1 + 11.9T + 43T^{2} \) |
| 47 | \( 1 + 10.1T + 47T^{2} \) |
| 53 | \( 1 + 7.07T + 53T^{2} \) |
| 59 | \( 1 + 14.5T + 59T^{2} \) |
| 61 | \( 1 - 0.515T + 61T^{2} \) |
| 67 | \( 1 + 3.40T + 67T^{2} \) |
| 71 | \( 1 - 10.7T + 71T^{2} \) |
| 73 | \( 1 - 2.74T + 73T^{2} \) |
| 79 | \( 1 - 6.98T + 79T^{2} \) |
| 83 | \( 1 + 6.03T + 83T^{2} \) |
| 89 | \( 1 + 3.61T + 89T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.79013692908020634661773593511, −7.58778496730762355774583966123, −6.69126397578134560015203068242, −5.40709574838230692233637823040, −4.90668359240275917022612308471, −3.91555440860883563894806061732, −3.33029718708829230915343307647, −2.91258532129007928264333844841, −1.70957076218156768916105278557, −0.02926928709506331555464250784,
0.02926928709506331555464250784, 1.70957076218156768916105278557, 2.91258532129007928264333844841, 3.33029718708829230915343307647, 3.91555440860883563894806061732, 4.90668359240275917022612308471, 5.40709574838230692233637823040, 6.69126397578134560015203068242, 7.58778496730762355774583966123, 7.79013692908020634661773593511
