| L(s) = 1 | + (1.26 − 0.633i)2-s + (1.19 − 1.60i)4-s + 0.133·5-s + 1.93i·7-s + (0.500 − 2.78i)8-s − 3·9-s + (0.168 − 0.0845i)10-s + (1.22 + 2.44i)14-s + (−1.13 − 3.83i)16-s + (−3.79 + 1.90i)18-s + 7.00i·19-s + (0.159 − 0.213i)20-s − 4.98·25-s + (3.10 + 2.31i)28-s − 5.56i·31-s + (−3.86 − 4.13i)32-s + ⋯ |
| L(s) = 1 | + (0.894 − 0.447i)2-s + (0.598 − 0.800i)4-s + 0.0596·5-s + 0.732i·7-s + (0.176 − 0.984i)8-s − 9-s + (0.0533 − 0.0267i)10-s + (0.327 + 0.654i)14-s + (−0.282 − 0.959i)16-s + (−0.894 + 0.447i)18-s + 1.60i·19-s + (0.0357 − 0.0478i)20-s − 0.996·25-s + (0.586 + 0.438i)28-s − 0.999i·31-s + (−0.682 − 0.730i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.800 + 0.598i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.800 + 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.53511 - 0.510486i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.53511 - 0.510486i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.26 + 0.633i)T \) |
| 31 | \( 1 + 5.56iT \) |
| good | 3 | \( 1 + 3T^{2} \) |
| 5 | \( 1 - 0.133T + 5T^{2} \) |
| 7 | \( 1 - 1.93iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 7.00iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 - 9.71T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 11.1iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 3.13iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 11.1iT - 67T^{2} \) |
| 71 | \( 1 + 15.9iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 + 9.44T + 97T^{2} \) |
| 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
−13.29942972413018874533739083258, −12.16126441884699010484821628151, −11.59122866507962943662338573120, −10.41087855860953119200851381786, −9.266340901342059839717506733728, −7.88303888171488413924403473268, −6.15084496139736414507903549523, −5.46131642298374612464632010154, −3.78960485947151438374689351693, −2.30348544053450484222815280526,
2.84140508060737160215985701772, 4.33239507926860768016469451683, 5.61411114973058225388825223302, 6.80757012245788040692269084572, 7.88181288886124596156311600610, 9.113915182989517999740553263343, 10.79339236945857453907547685368, 11.53529311194528264677427059465, 12.71275403013188988191917359000, 13.71858868978520258342468005809
