L(s) = 1 | + 4.33i·5-s + 2.30·7-s + 0.582i·11-s + i·13-s − 3.40·17-s + 0.0627i·19-s − 6.65·23-s − 13.7·25-s + 2.41i·29-s − 5.63·31-s + 9.99i·35-s − 9.27i·37-s − 4.75·41-s − 7.86i·43-s − 0.0312·47-s + ⋯ |
L(s) = 1 | + 1.93i·5-s + 0.870·7-s + 0.175i·11-s + 0.277i·13-s − 0.826·17-s + 0.0143i·19-s − 1.38·23-s − 2.75·25-s + 0.449i·29-s − 1.01·31-s + 1.68i·35-s − 1.52i·37-s − 0.742·41-s − 1.19i·43-s − 0.00456·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.937 + 0.348i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.937 + 0.348i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7703732611\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7703732611\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 - iT \) |
good | 5 | \( 1 - 4.33iT - 5T^{2} \) |
| 7 | \( 1 - 2.30T + 7T^{2} \) |
| 11 | \( 1 - 0.582iT - 11T^{2} \) |
| 17 | \( 1 + 3.40T + 17T^{2} \) |
| 19 | \( 1 - 0.0627iT - 19T^{2} \) |
| 23 | \( 1 + 6.65T + 23T^{2} \) |
| 29 | \( 1 - 2.41iT - 29T^{2} \) |
| 31 | \( 1 + 5.63T + 31T^{2} \) |
| 37 | \( 1 + 9.27iT - 37T^{2} \) |
| 41 | \( 1 + 4.75T + 41T^{2} \) |
| 43 | \( 1 + 7.86iT - 43T^{2} \) |
| 47 | \( 1 + 0.0312T + 47T^{2} \) |
| 53 | \( 1 - 13.1iT - 53T^{2} \) |
| 59 | \( 1 + 4.75iT - 59T^{2} \) |
| 61 | \( 1 + 3.29iT - 61T^{2} \) |
| 67 | \( 1 - 11.8iT - 67T^{2} \) |
| 71 | \( 1 - 13.6T + 71T^{2} \) |
| 73 | \( 1 - 0.437T + 73T^{2} \) |
| 79 | \( 1 - 6.54T + 79T^{2} \) |
| 83 | \( 1 + 6.62iT - 83T^{2} \) |
| 89 | \( 1 + 8.23T + 89T^{2} \) |
| 97 | \( 1 + 0.664T + 97T^{2} \) |
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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
−8.924466951202264550076093670176, −7.960431047324269667378156439716, −7.40086938670976811779730360982, −6.78874710039670402232361645550, −6.09929126418422180645113046639, −5.27386872895705688508251983676, −4.11618105475801430921255166830, −3.56255340331718686089668015693, −2.37888787200335244828841581861, −1.91844373632553967972450255268,
0.20974032814779323406137569778, 1.40395973759119636274084227421, 2.08820300934540457456764495417, 3.64094910110442832539838313630, 4.49813187448714883937104446790, 4.95570578699212947529825940078, 5.66311014956829408139917235931, 6.50327857977959143779550418322, 7.72615205504908012020745377884, 8.282977696094322910775975850778