L(s) = 1 | + 2.35·2-s + 3-s + 3.54·4-s + 5-s + 2.35·6-s − 0.193·7-s + 3.64·8-s + 9-s + 2.35·10-s + 3.54·12-s − 0.973·13-s − 0.455·14-s + 15-s + 1.49·16-s + 2.67·17-s + 2.35·18-s + 5.54·19-s + 3.54·20-s − 0.193·21-s − 4.80·23-s + 3.64·24-s + 25-s − 2.29·26-s + 27-s − 0.686·28-s + 10.1·29-s + 2.35·30-s + ⋯ |
L(s) = 1 | + 1.66·2-s + 0.577·3-s + 1.77·4-s + 0.447·5-s + 0.961·6-s − 0.0731·7-s + 1.29·8-s + 0.333·9-s + 0.744·10-s + 1.02·12-s − 0.270·13-s − 0.121·14-s + 0.258·15-s + 0.374·16-s + 0.648·17-s + 0.555·18-s + 1.27·19-s + 0.793·20-s − 0.0422·21-s − 1.00·23-s + 0.744·24-s + 0.200·25-s − 0.449·26-s + 0.192·27-s − 0.129·28-s + 1.87·29-s + 0.430·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(6.043062334\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.043062334\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - 2.35T + 2T^{2} \) |
| 7 | \( 1 + 0.193T + 7T^{2} \) |
| 13 | \( 1 + 0.973T + 13T^{2} \) |
| 17 | \( 1 - 2.67T + 17T^{2} \) |
| 19 | \( 1 - 5.54T + 19T^{2} \) |
| 23 | \( 1 + 4.80T + 23T^{2} \) |
| 29 | \( 1 - 10.1T + 29T^{2} \) |
| 31 | \( 1 + 2.50T + 31T^{2} \) |
| 37 | \( 1 - 5.71T + 37T^{2} \) |
| 41 | \( 1 - 8.27T + 41T^{2} \) |
| 43 | \( 1 + 5.11T + 43T^{2} \) |
| 47 | \( 1 + 10.8T + 47T^{2} \) |
| 53 | \( 1 + 9.28T + 53T^{2} \) |
| 59 | \( 1 + 11.2T + 59T^{2} \) |
| 61 | \( 1 + 1.20T + 61T^{2} \) |
| 67 | \( 1 - 3.25T + 67T^{2} \) |
| 71 | \( 1 - 5.99T + 71T^{2} \) |
| 73 | \( 1 + 3.30T + 73T^{2} \) |
| 79 | \( 1 + 10.1T + 79T^{2} \) |
| 83 | \( 1 + 8.31T + 83T^{2} \) |
| 89 | \( 1 - 7.34T + 89T^{2} \) |
| 97 | \( 1 + 15.8T + 97T^{2} \) |
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
−9.510405745265142590518942471297, −8.254137922185241416476797677281, −7.53359435945286096346273318145, −6.57535766079166193415333779326, −5.93871752012208241956761582382, −5.04677735323847615249620167287, −4.37624537970547017974853502395, −3.28438806573780059091644503585, −2.77833282591387082970962317041, −1.57109557227935181812235434246,
1.57109557227935181812235434246, 2.77833282591387082970962317041, 3.28438806573780059091644503585, 4.37624537970547017974853502395, 5.04677735323847615249620167287, 5.93871752012208241956761582382, 6.57535766079166193415333779326, 7.53359435945286096346273318145, 8.254137922185241416476797677281, 9.510405745265142590518942471297