| L(s) = 1 | − 2.16i·5-s − 9.11·7-s − 1.10i·11-s + 2.28·13-s + 19.2i·17-s + 4.35·19-s + 1.28i·23-s + 20.3·25-s + 37.0i·29-s + 31.3·31-s + 19.6i·35-s + 34.9·37-s + 43.5i·41-s + 0.553·43-s − 9.35i·47-s + ⋯ |
| L(s) = 1 | − 0.432i·5-s − 1.30·7-s − 0.100i·11-s + 0.175·13-s + 1.13i·17-s + 0.229·19-s + 0.0559i·23-s + 0.813·25-s + 1.27i·29-s + 1.01·31-s + 0.562i·35-s + 0.944·37-s + 1.06i·41-s + 0.0128·43-s − 0.198i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.277372050\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.277372050\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 - 4.35T \) |
| good | 5 | \( 1 + 2.16iT - 25T^{2} \) |
| 7 | \( 1 + 9.11T + 49T^{2} \) |
| 11 | \( 1 + 1.10iT - 121T^{2} \) |
| 13 | \( 1 - 2.28T + 169T^{2} \) |
| 17 | \( 1 - 19.2iT - 289T^{2} \) |
| 23 | \( 1 - 1.28iT - 529T^{2} \) |
| 29 | \( 1 - 37.0iT - 841T^{2} \) |
| 31 | \( 1 - 31.3T + 961T^{2} \) |
| 37 | \( 1 - 34.9T + 1.36e3T^{2} \) |
| 41 | \( 1 - 43.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 0.553T + 1.84e3T^{2} \) |
| 47 | \( 1 + 9.35iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 75.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 38.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 62.4T + 3.72e3T^{2} \) |
| 67 | \( 1 - 85.6T + 4.48e3T^{2} \) |
| 71 | \( 1 + 15.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 67.1T + 5.32e3T^{2} \) |
| 79 | \( 1 + 58.5T + 6.24e3T^{2} \) |
| 83 | \( 1 - 115. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 6.59iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 14.0T + 9.40e3T^{2} \) |
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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
−10.33118556841104867834410896235, −9.527697320997277551150712169672, −8.779262405612190657597714035295, −7.88316039554588672969935031418, −6.68292175542382966478021487273, −6.09999211673983379638782437901, −4.93474010274908544027794017507, −3.77414241343977802236440826356, −2.81376600500207418615330310219, −1.11665968961109240194198091715,
0.53613223488668531145845719864, 2.52723009891828486767925363507, 3.34655469943449665573292401591, 4.55592608107898268242146099344, 5.81484536642604363890690704010, 6.63701356538280576136145123204, 7.33392195854315690240125475088, 8.470445350785701232388700876610, 9.542399759547575017612591018594, 9.939528242158654401550755813256
