L(s) = 1 | + (0.541 + 1.30i)2-s + (1.30 − 1.13i)3-s + (−1.41 + 1.41i)4-s + (2.10 − 0.765i)5-s + (2.19 + 1.09i)6-s − 2.27·7-s + (−2.61 − 1.08i)8-s + (0.414 − 2.97i)9-s + (2.13 + 2.33i)10-s + 4.20i·11-s + (−0.239 + 3.45i)12-s − 3.21·13-s + (−1.23 − 2.97i)14-s + (1.87 − 3.38i)15-s − 4i·16-s + 1.53·17-s + ⋯ |
L(s) = 1 | + (0.382 + 0.923i)2-s + (0.754 − 0.656i)3-s + (−0.707 + 0.707i)4-s + (0.939 − 0.342i)5-s + (0.895 + 0.445i)6-s − 0.859·7-s + (−0.923 − 0.382i)8-s + (0.138 − 0.990i)9-s + (0.675 + 0.737i)10-s + 1.26i·11-s + (−0.0692 + 0.997i)12-s − 0.891·13-s + (−0.328 − 0.794i)14-s + (0.484 − 0.875i)15-s − i·16-s + 0.371·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.782 - 0.623i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.782 - 0.623i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.38754 + 0.485202i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.38754 + 0.485202i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.541 - 1.30i)T \) |
| 3 | \( 1 + (-1.30 + 1.13i)T \) |
| 5 | \( 1 + (-2.10 + 0.765i)T \) |
good | 7 | \( 1 + 2.27T + 7T^{2} \) |
| 11 | \( 1 - 4.20iT - 11T^{2} \) |
| 13 | \( 1 + 3.21T + 13T^{2} \) |
| 17 | \( 1 - 1.53T + 17T^{2} \) |
| 19 | \( 1 + 4.82T + 19T^{2} \) |
| 23 | \( 1 - 1.08iT - 23T^{2} \) |
| 29 | \( 1 - 1.74T + 29T^{2} \) |
| 31 | \( 1 + 6.82iT - 31T^{2} \) |
| 37 | \( 1 - 7.76T + 37T^{2} \) |
| 41 | \( 1 + 2.46iT - 41T^{2} \) |
| 43 | \( 1 - 8.70iT - 43T^{2} \) |
| 47 | \( 1 + 1.08iT - 47T^{2} \) |
| 53 | \( 1 - 11.0iT - 53T^{2} \) |
| 59 | \( 1 + 4.20iT - 59T^{2} \) |
| 61 | \( 1 - 8.48iT - 61T^{2} \) |
| 67 | \( 1 - 2.27iT - 67T^{2} \) |
| 71 | \( 1 - 11.8T + 71T^{2} \) |
| 73 | \( 1 + 4.54iT - 73T^{2} \) |
| 79 | \( 1 + 0.485iT - 79T^{2} \) |
| 83 | \( 1 - 6.94T + 83T^{2} \) |
| 89 | \( 1 - 8.40iT - 89T^{2} \) |
| 97 | \( 1 + 10.9iT - 97T^{2} \) |
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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
−13.56316757210808423489090171784, −12.79975771513327906528693383002, −12.32392081998281347490556967045, −9.791236946817159502717927280096, −9.356721776099003174038156149692, −7.986851897594678504804044574208, −6.92700848749489147962584129560, −6.01962328633680309308725454851, −4.41457162137658854128575715410, −2.55484236272690200111100406747,
2.48009833454209043096434564647, 3.50207224093443734710015805363, 5.12056728808378425870375499291, 6.39932684106204987704310751492, 8.495596996186060220843132942190, 9.480476106158081012508620652035, 10.20744571437767667737739833959, 11.01157329627151345804455378008, 12.58790517604808832053006694507, 13.42517704310427999401719126874