L(s) = 1 | + (0.707 − 0.707i)2-s + (−0.707 + 0.707i)3-s − 1.00i·4-s + (1.89 + 1.19i)5-s + 1.00i·6-s + (2.48 − 2.48i)7-s + (−0.707 − 0.707i)8-s − 1.00i·9-s + (2.18 − 0.496i)10-s − 3.08i·11-s + (0.707 + 0.707i)12-s + (−0.986 + 0.986i)13-s − 3.51i·14-s + (−2.18 + 0.496i)15-s − 1.00·16-s + (−1.61 − 1.61i)17-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s + (−0.408 + 0.408i)3-s − 0.500i·4-s + (0.846 + 0.532i)5-s + 0.408i·6-s + (0.938 − 0.938i)7-s + (−0.250 − 0.250i)8-s − 0.333i·9-s + (0.689 − 0.156i)10-s − 0.930i·11-s + (0.204 + 0.204i)12-s + (−0.273 + 0.273i)13-s − 0.938i·14-s + (−0.562 + 0.128i)15-s − 0.250·16-s + (−0.390 − 0.390i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.509 + 0.860i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.509 + 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.93175 - 1.10067i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.93175 - 1.10067i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.89 - 1.19i)T \) |
| 31 | \( 1 + (-0.442 - 5.55i)T \) |
good | 7 | \( 1 + (-2.48 + 2.48i)T - 7iT^{2} \) |
| 11 | \( 1 + 3.08iT - 11T^{2} \) |
| 13 | \( 1 + (0.986 - 0.986i)T - 13iT^{2} \) |
| 17 | \( 1 + (1.61 + 1.61i)T + 17iT^{2} \) |
| 19 | \( 1 + 2.13iT - 19T^{2} \) |
| 23 | \( 1 + (-2.98 + 2.98i)T - 23iT^{2} \) |
| 29 | \( 1 - 3.68T + 29T^{2} \) |
| 37 | \( 1 + (0.671 + 0.671i)T + 37iT^{2} \) |
| 41 | \( 1 + 2.47T + 41T^{2} \) |
| 43 | \( 1 + (-1.90 + 1.90i)T - 43iT^{2} \) |
| 47 | \( 1 + (0.635 - 0.635i)T - 47iT^{2} \) |
| 53 | \( 1 + (-4.01 + 4.01i)T - 53iT^{2} \) |
| 59 | \( 1 + 7.25iT - 59T^{2} \) |
| 61 | \( 1 - 1.31iT - 61T^{2} \) |
| 67 | \( 1 + (0.962 - 0.962i)T - 67iT^{2} \) |
| 71 | \( 1 - 9.05T + 71T^{2} \) |
| 73 | \( 1 + (2.24 - 2.24i)T - 73iT^{2} \) |
| 79 | \( 1 - 14.6T + 79T^{2} \) |
| 83 | \( 1 + (7.83 - 7.83i)T - 83iT^{2} \) |
| 89 | \( 1 - 0.895T + 89T^{2} \) |
| 97 | \( 1 + (3.59 - 3.59i)T - 97iT^{2} \) |
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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
−10.24953141688493919314786703754, −9.338087554582753851812175186133, −8.415188253655086775650933618414, −7.05990945601511796062566649129, −6.45143165489364550253824734972, −5.28306463160372536816871085636, −4.72015829694558448249971198097, −3.58236145578285707414587008908, −2.46164223269465360309560093716, −1.03817304991105830195518338861,
1.61665369075415974951331903866, 2.53755583012459903491890026443, 4.37075007564296764042850479748, 5.15481378135180442929033471568, 5.72875357334375087917020938404, 6.61233096017990662740965122050, 7.64136472425310855432210626459, 8.402897434712800711952909147148, 9.232800123438093960221978212276, 10.15476337204078311523926853265