L(s) = 1 | − 2-s + 1.76i·3-s + 4-s + (−1.62 − 1.53i)5-s − 1.76i·6-s − 1.22i·7-s − 8-s − 0.105·9-s + (1.62 + 1.53i)10-s + 1.87·11-s + 1.76i·12-s + 6.50·13-s + 1.22i·14-s + (2.69 − 2.86i)15-s + 16-s − 0.765·17-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.01i·3-s + 0.5·4-s + (−0.728 − 0.685i)5-s − 0.719i·6-s − 0.461i·7-s − 0.353·8-s − 0.0350·9-s + (0.515 + 0.484i)10-s + 0.564·11-s + 0.508i·12-s + 1.80·13-s + 0.326i·14-s + (0.697 − 0.741i)15-s + 0.250·16-s − 0.185·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.867 - 0.498i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.867 - 0.498i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.941595 + 0.251173i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.941595 + 0.251173i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 5 | \( 1 + (1.62 + 1.53i)T \) |
| 37 | \( 1 + (-1.76 + 5.82i)T \) |
good | 3 | \( 1 - 1.76iT - 3T^{2} \) |
| 7 | \( 1 + 1.22iT - 7T^{2} \) |
| 11 | \( 1 - 1.87T + 11T^{2} \) |
| 13 | \( 1 - 6.50T + 13T^{2} \) |
| 17 | \( 1 + 0.765T + 17T^{2} \) |
| 19 | \( 1 - 3.34iT - 19T^{2} \) |
| 23 | \( 1 - 1.38T + 23T^{2} \) |
| 29 | \( 1 + 1.72iT - 29T^{2} \) |
| 31 | \( 1 - 4.11iT - 31T^{2} \) |
| 41 | \( 1 - 3.73T + 41T^{2} \) |
| 43 | \( 1 - 4.91T + 43T^{2} \) |
| 47 | \( 1 + 6.30iT - 47T^{2} \) |
| 53 | \( 1 + 2.57iT - 53T^{2} \) |
| 59 | \( 1 - 10.5iT - 59T^{2} \) |
| 61 | \( 1 + 11.1iT - 61T^{2} \) |
| 67 | \( 1 + 11.1iT - 67T^{2} \) |
| 71 | \( 1 - 0.963T + 71T^{2} \) |
| 73 | \( 1 - 9.03iT - 73T^{2} \) |
| 79 | \( 1 - 10.3iT - 79T^{2} \) |
| 83 | \( 1 + 0.00656iT - 83T^{2} \) |
| 89 | \( 1 + 4.70iT - 89T^{2} \) |
| 97 | \( 1 + 0.403T + 97T^{2} \) |
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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
−11.07824747886869337902474607475, −10.65319467326268095857746685454, −9.504174087907260037187396261016, −8.840185474484417401951715016932, −8.031938325251895099260035430879, −6.88535761797039082456678625769, −5.60169504965734155355167032167, −4.17905450228621559789044926836, −3.64250923049939807967731791641, −1.20856738834502712407637389645,
1.15202431365447672837557696624, 2.68347517565279136380534429680, 4.05493135743222481555777632216, 6.07322218884136776963966084042, 6.69890240232247767175370520542, 7.58756830587059274868569191608, 8.420926461727632493373473624607, 9.255633442883291945617117408796, 10.62587184559076903945748741297, 11.32781275735716117572915677398