L(s) = 1 | + 0.193i·2-s + i·3-s + 1.96·4-s − 0.193·6-s − 3.35i·7-s + 0.768i·8-s − 9-s + 11-s + 1.96i·12-s + 2.96i·13-s + 0.649·14-s + 3.77·16-s + 4.57i·17-s − 0.193i·18-s + 4.31·19-s + ⋯ |
L(s) = 1 | + 0.137i·2-s + 0.577i·3-s + 0.981·4-s − 0.0791·6-s − 1.26i·7-s + 0.271i·8-s − 0.333·9-s + 0.301·11-s + 0.566i·12-s + 0.821i·13-s + 0.173·14-s + 0.943·16-s + 1.10i·17-s − 0.0457i·18-s + 0.989·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.98673 + 0.469005i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.98673 + 0.469005i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 2 | \( 1 - 0.193iT - 2T^{2} \) |
| 7 | \( 1 + 3.35iT - 7T^{2} \) |
| 13 | \( 1 - 2.96iT - 13T^{2} \) |
| 17 | \( 1 - 4.57iT - 17T^{2} \) |
| 19 | \( 1 - 4.31T + 19T^{2} \) |
| 23 | \( 1 + 6.70iT - 23T^{2} \) |
| 29 | \( 1 - 3.61T + 29T^{2} \) |
| 31 | \( 1 - 9.92T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 + 4.38T + 41T^{2} \) |
| 43 | \( 1 + 9.27iT - 43T^{2} \) |
| 47 | \( 1 - 9.92iT - 47T^{2} \) |
| 53 | \( 1 - 4.70iT - 53T^{2} \) |
| 59 | \( 1 + 10.7T + 59T^{2} \) |
| 61 | \( 1 + 8.70T + 61T^{2} \) |
| 67 | \( 1 + 5.92iT - 67T^{2} \) |
| 71 | \( 1 - 9.92T + 71T^{2} \) |
| 73 | \( 1 + 7.73iT - 73T^{2} \) |
| 79 | \( 1 + 11.5T + 79T^{2} \) |
| 83 | \( 1 - 10.8iT - 83T^{2} \) |
| 89 | \( 1 - 2.77T + 89T^{2} \) |
| 97 | \( 1 + 0.0752iT - 97T^{2} \) |
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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
−10.49444850563457452080787149885, −9.653517278852827041322262666276, −8.487405890246587390333854149178, −7.68201168116726889977207457736, −6.70717233275287770241856914797, −6.19613021581127704028967135137, −4.76248189017050633318029523059, −3.94791924414931075329544765749, −2.83793424940657830106505447380, −1.31990427721858131701545632652,
1.28445986406779272986953938203, 2.60164682178265468323678504723, 3.19805980919521265372365458022, 5.11637270973882560114949791046, 5.84257126019752337364826620180, 6.69987743046074243863340713950, 7.57891468013322845268041489155, 8.308424888568640151846615893969, 9.391347784781869177756618823199, 10.11880021871734452864711563803