| L(s) = 1 | − i·2-s + 3-s − 4-s − i·5-s − i·6-s − 2i·7-s + i·8-s + 9-s − 10-s + 3i·11-s − 12-s − 2·14-s − i·15-s + 16-s − 6·17-s − i·18-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.577·3-s − 0.5·4-s − 0.447i·5-s − 0.408i·6-s − 0.755i·7-s + 0.353i·8-s + 0.333·9-s − 0.316·10-s + 0.904i·11-s − 0.288·12-s − 0.534·14-s − 0.258i·15-s + 0.250·16-s − 1.45·17-s − 0.235i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.326633520\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.326633520\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + iT \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 - 3iT - 11T^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 - 2iT - 19T^{2} \) |
| 23 | \( 1 + 3T + 23T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 - 5iT - 31T^{2} \) |
| 37 | \( 1 - 7iT - 37T^{2} \) |
| 41 | \( 1 - 6iT - 41T^{2} \) |
| 43 | \( 1 - T + 43T^{2} \) |
| 47 | \( 1 - 3iT - 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 - 9iT - 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 8iT - 67T^{2} \) |
| 71 | \( 1 + 12iT - 71T^{2} \) |
| 73 | \( 1 + 14iT - 73T^{2} \) |
| 79 | \( 1 - 5T + 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 - 18iT - 89T^{2} \) |
| 97 | \( 1 - 14iT - 97T^{2} \) |
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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
−8.326249008305314558683950913306, −7.79554238499848118647564076581, −6.92915391690841619223447171287, −6.25230391679810651195193028337, −4.95852894655401897808059470391, −4.47503395124674126951283021363, −3.84959798874664381953104334032, −2.86257019471197360452941215465, −1.97452111022205197322206413750, −1.15658001409719249031054745265,
0.33373083231994877022887768936, 2.04866679336458395753259929269, 2.73372979041411725114475866718, 3.72825288040742446441310192257, 4.42947053621484210932482280760, 5.44939507567546335458426373303, 6.05740187538049644759979417788, 6.76488647011071174747610385004, 7.39853799603499075590438023952, 8.349542012464486739397088149720
