L(s) = 1 | + (−0.707 + 0.707i)2-s − 1.00i·4-s + (0.707 + 0.707i)7-s + (0.707 + 0.707i)8-s + 0.585i·11-s + (3.43 − 3.43i)13-s − 1.00·14-s − 1.00·16-s + (0.906 − 0.906i)17-s − 2.04i·19-s + (−0.414 − 0.414i)22-s + (−0.257 − 0.257i)23-s + 4.86i·26-s + (0.707 − 0.707i)28-s − 1.75·29-s + ⋯ |
L(s) = 1 | + (−0.499 + 0.499i)2-s − 0.500i·4-s + (0.267 + 0.267i)7-s + (0.250 + 0.250i)8-s + 0.176i·11-s + (0.953 − 0.953i)13-s − 0.267·14-s − 0.250·16-s + (0.219 − 0.219i)17-s − 0.470i·19-s + (−0.0883 − 0.0883i)22-s + (−0.0537 − 0.0537i)23-s + 0.953i·26-s + (0.133 − 0.133i)28-s − 0.325·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.876 + 0.481i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.876 + 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.355407118\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.355407118\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 - 0.707i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.707 - 0.707i)T \) |
good | 11 | \( 1 - 0.585iT - 11T^{2} \) |
| 13 | \( 1 + (-3.43 + 3.43i)T - 13iT^{2} \) |
| 17 | \( 1 + (-0.906 + 0.906i)T - 17iT^{2} \) |
| 19 | \( 1 + 2.04iT - 19T^{2} \) |
| 23 | \( 1 + (0.257 + 0.257i)T + 23iT^{2} \) |
| 29 | \( 1 + 1.75T + 29T^{2} \) |
| 31 | \( 1 - 1.47T + 31T^{2} \) |
| 37 | \( 1 + (4.04 + 4.04i)T + 37iT^{2} \) |
| 41 | \( 1 - 3.84iT - 41T^{2} \) |
| 43 | \( 1 + (-5.10 + 5.10i)T - 43iT^{2} \) |
| 47 | \( 1 + (-5.49 + 5.49i)T - 47iT^{2} \) |
| 53 | \( 1 + (5.02 + 5.02i)T + 53iT^{2} \) |
| 59 | \( 1 + 11.1T + 59T^{2} \) |
| 61 | \( 1 - 3.78T + 61T^{2} \) |
| 67 | \( 1 + (6.74 + 6.74i)T + 67iT^{2} \) |
| 71 | \( 1 - 10.9iT - 71T^{2} \) |
| 73 | \( 1 + (-5.97 + 5.97i)T - 73iT^{2} \) |
| 79 | \( 1 + 0.944iT - 79T^{2} \) |
| 83 | \( 1 + (-7.82 - 7.82i)T + 83iT^{2} \) |
| 89 | \( 1 + 0.0705T + 89T^{2} \) |
| 97 | \( 1 + (0.746 + 0.746i)T + 97iT^{2} \) |
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
−8.583968632306023701349787995888, −7.914534388488854641669554002752, −7.24945912299673001324745627628, −6.38451393259366439357783338601, −5.65174788981172561628817232307, −5.01866075136280929340563663509, −3.94457034206511337684266411679, −2.92823324811952693780306745833, −1.77938070022417431632864397314, −0.57858624825547086064757433847,
1.09667169838984893080014557780, 1.92167938491223065497947516093, 3.12282890035675066197319588295, 3.93971630961173754015594192193, 4.66191202924274194107907913257, 5.84878328782956041148614810615, 6.50730860863598160918032917053, 7.46718117009401462822416718728, 8.056846526338130419998247373474, 8.883861617855262072476257259017