| L(s) = 1 | − 2i·2-s − 2·4-s − i·7-s − 5·11-s + i·13-s − 2·14-s − 4·16-s + 7i·17-s + 6·19-s + 10i·22-s + 3i·23-s + 2·26-s + 2i·28-s + 2·29-s + 2·31-s + 8i·32-s + ⋯ |
| L(s) = 1 | − 1.41i·2-s − 4-s − 0.377i·7-s − 1.50·11-s + 0.277i·13-s − 0.534·14-s − 16-s + 1.69i·17-s + 1.37·19-s + 2.13i·22-s + 0.625i·23-s + 0.392·26-s + 0.377i·28-s + 0.371·29-s + 0.359·31-s + 1.41i·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
| \(L(1)\) |
\(\approx\) |
\(1.179470486\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.179470486\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2iT - 2T^{2} \) |
| 7 | \( 1 + iT - 7T^{2} \) |
| 11 | \( 1 + 5T + 11T^{2} \) |
| 17 | \( 1 - 7iT - 17T^{2} \) |
| 19 | \( 1 - 6T + 19T^{2} \) |
| 23 | \( 1 - 3iT - 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 - 7iT - 37T^{2} \) |
| 41 | \( 1 + 9T + 41T^{2} \) |
| 43 | \( 1 - 8iT - 43T^{2} \) |
| 47 | \( 1 + 10iT - 47T^{2} \) |
| 53 | \( 1 - 5iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 5T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 + 9T + 71T^{2} \) |
| 73 | \( 1 - 6iT - 73T^{2} \) |
| 79 | \( 1 - 3T + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 - 11T + 89T^{2} \) |
| 97 | \( 1 + 11iT - 97T^{2} \) |
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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
−8.829263857119788647298578999098, −8.092596354693305673925359911071, −7.34346829919177236668793405576, −6.39637735725853034284312317121, −5.37876792662062960504528396697, −4.60104546697356809633934949749, −3.62232136451929325336670008464, −3.01918928634605029832731920160, −2.01034850016492351894221002289, −1.07835761842625531235173049725,
0.41274208485020410182178715212, 2.37046312943566954663974412878, 3.08186546067727107830046240922, 4.57761421465896162074233549631, 5.33833862112796163793126472740, 5.55212582984307859010543223628, 6.69535598486770611679807880819, 7.35204717627904115224397706269, 7.82487302143149079601235297046, 8.586103621581181900725593975737
