| L(s) = 1 | + 2.41i·2-s − 3.82·4-s + 0.828i·7-s − 4.41i·8-s + 11-s + 5.65i·13-s − 1.99·14-s + 2.99·16-s + 1.17i·17-s + 6.82·19-s + 2.41i·22-s − 4i·23-s − 13.6·26-s − 3.17i·28-s − 4.82·29-s + ⋯ |
| L(s) = 1 | + 1.70i·2-s − 1.91·4-s + 0.313i·7-s − 1.56i·8-s + 0.301·11-s + 1.56i·13-s − 0.534·14-s + 0.749·16-s + 0.284i·17-s + 1.56·19-s + 0.514i·22-s − 0.834i·23-s − 2.67·26-s − 0.599i·28-s − 0.896·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.142545298\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.142545298\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 - 2.41iT - 2T^{2} \) |
| 7 | \( 1 - 0.828iT - 7T^{2} \) |
| 13 | \( 1 - 5.65iT - 13T^{2} \) |
| 17 | \( 1 - 1.17iT - 17T^{2} \) |
| 19 | \( 1 - 6.82T + 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 + 4.82T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 11.6iT - 37T^{2} \) |
| 41 | \( 1 + 4.82T + 41T^{2} \) |
| 43 | \( 1 - 8.82iT - 43T^{2} \) |
| 47 | \( 1 - 4iT - 47T^{2} \) |
| 53 | \( 1 - 9.31iT - 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 11.6T + 61T^{2} \) |
| 67 | \( 1 + 5.65iT - 67T^{2} \) |
| 71 | \( 1 + 2.34T + 71T^{2} \) |
| 73 | \( 1 + 11.3iT - 73T^{2} \) |
| 79 | \( 1 + 8.48T + 79T^{2} \) |
| 83 | \( 1 + 10iT - 83T^{2} \) |
| 89 | \( 1 - 3.65T + 89T^{2} \) |
| 97 | \( 1 - 11.6iT - 97T^{2} \) |
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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
−9.289395883510053614591950561743, −8.531240571515750983246546513408, −7.75809995609472801727269168882, −7.09601869756324309736631788315, −6.38783582181425935419151937416, −5.84624012248208243183898944078, −4.81682428111886250463378931661, −4.33552617551814117429856230038, −3.08515366300691333099474903900, −1.52319634873200051154830910074,
0.41187736469541232322826991742, 1.39794245395925405540891681939, 2.53329294135711272127591792732, 3.44076808623735226929035088474, 3.88378118846950798331584440067, 5.19714322992353404529127091013, 5.59249373788849695434543582942, 7.13384223044731206270244568568, 7.73626729568590850113583674432, 8.786629619200477560483591305558
