L(s) = 1 | − i·2-s − 4-s + 2i·7-s + i·8-s − 4·11-s + i·13-s + 2·14-s + 16-s − 8i·17-s + 6·19-s + 4i·22-s + 6i·23-s + 26-s − 2i·28-s − 4·29-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.5·4-s + 0.755i·7-s + 0.353i·8-s − 1.20·11-s + 0.277i·13-s + 0.534·14-s + 0.250·16-s − 1.94i·17-s + 1.37·19-s + 0.852i·22-s + 1.25i·23-s + 0.196·26-s − 0.377i·28-s − 0.742·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9535994616\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9535994616\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - iT \) |
good | 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 17 | \( 1 + 8iT - 17T^{2} \) |
| 19 | \( 1 - 6T + 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 + 4T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 + 10iT - 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 - 12iT - 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 - 8iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 + 14T + 89T^{2} \) |
| 97 | \( 1 + 16iT - 97T^{2} \) |
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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
−8.284409498918478212573024667001, −7.50633066083979316107929889418, −7.07099076656885750425054321859, −5.65148674972398163299322406546, −5.39842198843960328438765979144, −4.69255021877772115978277856029, −3.54823700787849834052437616517, −2.83510649403262469451582481210, −2.23784057413232239015662363670, −1.01228458345248967313177461384,
0.28063209462906563364628391652, 1.50854464504898687054929078518, 2.76715665954371806895572288870, 3.67448519944382708422263868417, 4.39246975800362600192230329504, 5.21823898635676650945669841281, 5.87034818797974510518008619838, 6.53653017829304975585248196308, 7.49583382444940258429579924327, 7.76724301066102644703893092073