| L(s) = 1 | − 0.447·2-s + 2.52·3-s − 1.80·4-s − 1.13·6-s − 3.76·7-s + 1.69·8-s + 3.40·9-s + 4.61·11-s − 4.55·12-s − 3.87·13-s + 1.68·14-s + 2.84·16-s + 3.87·17-s − 1.52·18-s − 5.29·19-s − 9.53·21-s − 2.06·22-s + 9.04·23-s + 4.29·24-s + 1.73·26-s + 1.01·27-s + 6.78·28-s − 4.06·29-s + 3.52·31-s − 4.66·32-s + 11.6·33-s − 1.73·34-s + ⋯ |
| L(s) = 1 | − 0.316·2-s + 1.46·3-s − 0.900·4-s − 0.461·6-s − 1.42·7-s + 0.600·8-s + 1.13·9-s + 1.39·11-s − 1.31·12-s − 1.07·13-s + 0.450·14-s + 0.710·16-s + 0.940·17-s − 0.358·18-s − 1.21·19-s − 2.08·21-s − 0.440·22-s + 1.88·23-s + 0.877·24-s + 0.339·26-s + 0.195·27-s + 1.28·28-s − 0.754·29-s + 0.632·31-s − 0.825·32-s + 2.03·33-s − 0.297·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.898356024\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.898356024\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 241 | \( 1 + T \) |
| good | 2 | \( 1 + 0.447T + 2T^{2} \) |
| 3 | \( 1 - 2.52T + 3T^{2} \) |
| 7 | \( 1 + 3.76T + 7T^{2} \) |
| 11 | \( 1 - 4.61T + 11T^{2} \) |
| 13 | \( 1 + 3.87T + 13T^{2} \) |
| 17 | \( 1 - 3.87T + 17T^{2} \) |
| 19 | \( 1 + 5.29T + 19T^{2} \) |
| 23 | \( 1 - 9.04T + 23T^{2} \) |
| 29 | \( 1 + 4.06T + 29T^{2} \) |
| 31 | \( 1 - 3.52T + 31T^{2} \) |
| 37 | \( 1 - 8.42T + 37T^{2} \) |
| 41 | \( 1 + 2.64T + 41T^{2} \) |
| 43 | \( 1 + 7.38T + 43T^{2} \) |
| 47 | \( 1 + 5.04T + 47T^{2} \) |
| 53 | \( 1 - 3.84T + 53T^{2} \) |
| 59 | \( 1 + 1.39T + 59T^{2} \) |
| 61 | \( 1 + 7.15T + 61T^{2} \) |
| 67 | \( 1 + 7.13T + 67T^{2} \) |
| 71 | \( 1 - 13.8T + 71T^{2} \) |
| 73 | \( 1 - 7.83T + 73T^{2} \) |
| 79 | \( 1 - 2.00T + 79T^{2} \) |
| 83 | \( 1 - 3.77T + 83T^{2} \) |
| 89 | \( 1 + 0.414T + 89T^{2} \) |
| 97 | \( 1 - 1.63T + 97T^{2} \) |
| 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
−8.242822266754461849472608338219, −7.52622938935016613409702269744, −6.84695147400422606382891909065, −6.14691156191248921590658978353, −4.97946355572921172316854856539, −4.22115675505661614788799642777, −3.43342164025692152142259946525, −3.04895548517185685882705738018, −1.91568335301722865013605168612, −0.71197223732762290003513582600,
0.71197223732762290003513582600, 1.91568335301722865013605168612, 3.04895548517185685882705738018, 3.43342164025692152142259946525, 4.22115675505661614788799642777, 4.97946355572921172316854856539, 6.14691156191248921590658978353, 6.84695147400422606382891909065, 7.52622938935016613409702269744, 8.242822266754461849472608338219
