L(s) = 1 | + (0.599 − 1.28i)2-s + 0.936i·3-s + (−1.28 − 1.53i)4-s + (1.19 + 0.561i)6-s + (−0.468 + 2.60i)7-s + (−2.73 + 0.719i)8-s + 2.12·9-s − 2.39i·11-s + (1.43 − 1.19i)12-s + 2·13-s + (3.05 + 2.16i)14-s + (−0.719 + 3.93i)16-s + 7.12·17-s + (1.27 − 2.71i)18-s − 2.39·19-s + ⋯ |
L(s) = 1 | + (0.424 − 0.905i)2-s + 0.540i·3-s + (−0.640 − 0.768i)4-s + (0.489 + 0.229i)6-s + (−0.176 + 0.984i)7-s + (−0.967 + 0.254i)8-s + 0.707·9-s − 0.723i·11-s + (0.415 − 0.346i)12-s + 0.554·13-s + (0.816 + 0.577i)14-s + (−0.179 + 0.983i)16-s + 1.72·17-s + (0.300 − 0.640i)18-s − 0.550·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.830 + 0.556i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.830 + 0.556i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.79905 - 0.546722i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.79905 - 0.546722i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.599 + 1.28i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (0.468 - 2.60i)T \) |
good | 3 | \( 1 - 0.936iT - 3T^{2} \) |
| 11 | \( 1 + 2.39iT - 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 - 7.12T + 17T^{2} \) |
| 19 | \( 1 + 2.39T + 19T^{2} \) |
| 23 | \( 1 - 5.73T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 6.67T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 + 7.12iT - 41T^{2} \) |
| 43 | \( 1 + 7.60T + 43T^{2} \) |
| 47 | \( 1 - 10.0iT - 47T^{2} \) |
| 53 | \( 1 - 2iT - 53T^{2} \) |
| 59 | \( 1 + 10.9T + 59T^{2} \) |
| 61 | \( 1 + 2iT - 61T^{2} \) |
| 67 | \( 1 - 14.2T + 67T^{2} \) |
| 71 | \( 1 + 6.14iT - 71T^{2} \) |
| 73 | \( 1 + 9.36T + 73T^{2} \) |
| 79 | \( 1 + 4.27iT - 79T^{2} \) |
| 83 | \( 1 - 0.936iT - 83T^{2} \) |
| 89 | \( 1 - 12iT - 89T^{2} \) |
| 97 | \( 1 + 7.12T + 97T^{2} \) |
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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
−10.46005694908690858923872521602, −9.648125190120449304630451016432, −8.967373134075996908462289240956, −8.076774375821704740000049946531, −6.49553972608357845787707996214, −5.60561176083287259447212288402, −4.80380557974312820641480270736, −3.63175580512690876885169275007, −2.85675365438659024879971898465, −1.27721607531681942341129047482,
1.16409457387159883069739887959, 3.17476491606323765463608195784, 4.21530023195643504068210216895, 5.08292083633555256138750149164, 6.39351272821206985877217851980, 6.94772378170229437301420233569, 7.70851546191513258179037631320, 8.413728558438311504483106777557, 9.753173803121566206999763606031, 10.24508598978273084359893787905