L(s) = 1 | − i·2-s + 4-s − 2i·7-s − 3i·8-s + 4·11-s − i·13-s − 2·14-s − 16-s − 4i·17-s − 6·19-s − 4i·22-s − 26-s − 2i·28-s − 4·29-s − 10·31-s − 5i·32-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + 0.5·4-s − 0.755i·7-s − 1.06i·8-s + 1.20·11-s − 0.277i·13-s − 0.534·14-s − 0.250·16-s − 0.970i·17-s − 1.37·19-s − 0.852i·22-s − 0.196·26-s − 0.377i·28-s − 0.742·29-s − 1.79·31-s − 0.883i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.928467372\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.928467372\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + iT \) |
good | 2 | \( 1 + iT - 2T^{2} \) |
| 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 17 | \( 1 + 4iT - 17T^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 4T + 29T^{2} \) |
| 31 | \( 1 + 10T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 8iT - 43T^{2} \) |
| 47 | \( 1 + 8iT - 47T^{2} \) |
| 53 | \( 1 - 4iT - 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 10iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 - 14T + 89T^{2} \) |
| 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
−8.640478354764349830338426811408, −7.38190372289339276174273781044, −7.07784595736577217867491647207, −6.28199089012103472046539940971, −5.34320139178100409240757325305, −3.99125849550683717147330295451, −3.81546704977612050231870385336, −2.55928115053030425957022667095, −1.67968237057189886127633171735, −0.56667796819195692297450569715,
1.64514646517185367173134214557, 2.33229378177938181478096311341, 3.61091765668983068671578109101, 4.43370217421671038091236394715, 5.61295205423045792346346692628, 6.07874853784744423193277160814, 6.71751492774084447385584828445, 7.47341377614696586697224382353, 8.330534744689757208337144450049, 8.904400100217382714915692269242