L(s) = 1 | + (−1.56 + 1.56i)2-s − 2.91i·4-s + (2.12 − 0.708i)5-s + (1.66 + 1.66i)7-s + (1.42 + 1.42i)8-s + (−2.21 + 4.43i)10-s + 6.00i·11-s + (0.950 − 0.950i)13-s − 5.21·14-s + 1.35·16-s + (−3.39 + 3.39i)17-s − 4.77i·19-s + (−2.06 − 6.17i)20-s + (−9.41 − 9.41i)22-s + (−0.707 − 0.707i)23-s + ⋯ |
L(s) = 1 | + (−1.10 + 1.10i)2-s − 1.45i·4-s + (0.948 − 0.317i)5-s + (0.629 + 0.629i)7-s + (0.504 + 0.504i)8-s + (−0.699 + 1.40i)10-s + 1.81i·11-s + (0.263 − 0.263i)13-s − 1.39·14-s + 0.337·16-s + (−0.822 + 0.822i)17-s − 1.09i·19-s + (−0.461 − 1.38i)20-s + (−2.00 − 2.00i)22-s + (−0.147 − 0.147i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.648 - 0.760i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.648 - 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.048175450\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.048175450\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (-2.12 + 0.708i)T \) |
| 23 | \( 1 + (0.707 + 0.707i)T \) |
good | 2 | \( 1 + (1.56 - 1.56i)T - 2iT^{2} \) |
| 7 | \( 1 + (-1.66 - 1.66i)T + 7iT^{2} \) |
| 11 | \( 1 - 6.00iT - 11T^{2} \) |
| 13 | \( 1 + (-0.950 + 0.950i)T - 13iT^{2} \) |
| 17 | \( 1 + (3.39 - 3.39i)T - 17iT^{2} \) |
| 19 | \( 1 + 4.77iT - 19T^{2} \) |
| 29 | \( 1 - 8.39T + 29T^{2} \) |
| 31 | \( 1 + 6.12T + 31T^{2} \) |
| 37 | \( 1 + (-8.08 - 8.08i)T + 37iT^{2} \) |
| 41 | \( 1 - 3.39iT - 41T^{2} \) |
| 43 | \( 1 + (3.81 - 3.81i)T - 43iT^{2} \) |
| 47 | \( 1 + (4.67 - 4.67i)T - 47iT^{2} \) |
| 53 | \( 1 + (1.97 + 1.97i)T + 53iT^{2} \) |
| 59 | \( 1 + 0.951T + 59T^{2} \) |
| 61 | \( 1 - 13.6T + 61T^{2} \) |
| 67 | \( 1 + (-4.40 - 4.40i)T + 67iT^{2} \) |
| 71 | \( 1 + 16.3iT - 71T^{2} \) |
| 73 | \( 1 + (8.01 - 8.01i)T - 73iT^{2} \) |
| 79 | \( 1 - 6.34iT - 79T^{2} \) |
| 83 | \( 1 + (-3.16 - 3.16i)T + 83iT^{2} \) |
| 89 | \( 1 - 4.16T + 89T^{2} \) |
| 97 | \( 1 + (3.80 + 3.80i)T + 97iT^{2} \) |
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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
−9.847641299767092689801036965310, −9.313471891777265873816425982694, −8.512238361104523633165163052990, −7.943684239658471953102071517122, −6.76200842718336533126568325992, −6.38886235914673417009914752677, −5.21913686307084769599027269568, −4.58780715370608369982491976292, −2.44837703674984453101648843514, −1.40396990230969397075914408957,
0.74576606723174692632407355376, 1.85010472639249095134889638891, 2.89608168512976924795441826419, 3.88818449887006303302467020391, 5.39646974816741975109437928078, 6.25641459232078388814980532968, 7.39961145044142019544411221829, 8.408185774068235479306448319411, 8.876953602116773702064384504490, 9.782087674192755702747321995712