L(s) = 1 | + (0.707 − 0.707i)2-s + (0.707 − 0.707i)3-s − 1.00i·4-s + (−1.96 + 1.06i)5-s − 1.00i·6-s + (−1.95 + 1.95i)7-s + (−0.707 − 0.707i)8-s − 1.00i·9-s + (−0.633 + 2.14i)10-s + 0.865i·11-s + (−0.707 − 0.707i)12-s + (−3.63 + 3.63i)13-s + 2.76i·14-s + (−0.633 + 2.14i)15-s − 1.00·16-s + (2.10 + 2.10i)17-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s + (0.408 − 0.408i)3-s − 0.500i·4-s + (−0.878 + 0.477i)5-s − 0.408i·6-s + (−0.738 + 0.738i)7-s + (−0.250 − 0.250i)8-s − 0.333i·9-s + (−0.200 + 0.678i)10-s + 0.260i·11-s + (−0.204 − 0.204i)12-s + (−1.00 + 1.00i)13-s + 0.738i·14-s + (−0.163 + 0.553i)15-s − 0.250·16-s + (0.510 + 0.510i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.111 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.111 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.686878 + 0.613886i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.686878 + 0.613886i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.707 + 0.707i)T \) |
| 5 | \( 1 + (1.96 - 1.06i)T \) |
| 31 | \( 1 + (-5.55 - 0.316i)T \) |
good | 7 | \( 1 + (1.95 - 1.95i)T - 7iT^{2} \) |
| 11 | \( 1 - 0.865iT - 11T^{2} \) |
| 13 | \( 1 + (3.63 - 3.63i)T - 13iT^{2} \) |
| 17 | \( 1 + (-2.10 - 2.10i)T + 17iT^{2} \) |
| 19 | \( 1 + 0.565iT - 19T^{2} \) |
| 23 | \( 1 + (6.50 - 6.50i)T - 23iT^{2} \) |
| 29 | \( 1 - 0.232T + 29T^{2} \) |
| 37 | \( 1 + (-2.57 - 2.57i)T + 37iT^{2} \) |
| 41 | \( 1 + 11.3T + 41T^{2} \) |
| 43 | \( 1 + (4.95 - 4.95i)T - 43iT^{2} \) |
| 47 | \( 1 + (-4.14 + 4.14i)T - 47iT^{2} \) |
| 53 | \( 1 + (0.513 - 0.513i)T - 53iT^{2} \) |
| 59 | \( 1 + 4.60iT - 59T^{2} \) |
| 61 | \( 1 + 2.89iT - 61T^{2} \) |
| 67 | \( 1 + (1.09 - 1.09i)T - 67iT^{2} \) |
| 71 | \( 1 + 9.81T + 71T^{2} \) |
| 73 | \( 1 + (0.542 - 0.542i)T - 73iT^{2} \) |
| 79 | \( 1 - 9.17T + 79T^{2} \) |
| 83 | \( 1 + (0.433 - 0.433i)T - 83iT^{2} \) |
| 89 | \( 1 + 3.35T + 89T^{2} \) |
| 97 | \( 1 + (-6.46 + 6.46i)T - 97iT^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.07997845756361762662927974552, −9.677419496066226351550765231654, −8.579771423084285250176320471562, −7.68394038043644627573547282180, −6.80766721086192130826737000691, −6.05480598039755779560574560368, −4.78334782862415576726589916463, −3.74012849499296630684874805751, −2.93138363320087238702226059298, −1.88566190661525682970387243129,
0.33099784034745582870538124327, 2.78432204362839721742954737092, 3.65392952640693688373060017399, 4.47273238252540904297691615614, 5.31697957900426825558833890333, 6.50223669425325514449856350992, 7.47435907801265371250131417216, 8.029339042319289617859559432989, 8.837979201190901945940094379927, 10.01826332573983948145409589757