L(s) = 1 | − i·2-s − i·3-s + 4-s + (−1 + 2i)5-s − 6-s − 4i·7-s − 3i·8-s − 9-s + (2 + i)10-s − i·12-s + 2i·13-s − 4·14-s + (2 + i)15-s − 16-s − 2i·17-s + i·18-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.577i·3-s + 0.5·4-s + (−0.447 + 0.894i)5-s − 0.408·6-s − 1.51i·7-s − 1.06i·8-s − 0.333·9-s + (0.632 + 0.316i)10-s − 0.288i·12-s + 0.554i·13-s − 1.06·14-s + (0.516 + 0.258i)15-s − 0.250·16-s − 0.485i·17-s + 0.235i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9584610083\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9584610083\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + iT \) |
| 5 | \( 1 + (1 - 2i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + iT - 2T^{2} \) |
| 7 | \( 1 + 4iT - 7T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 + 8T + 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 + 4T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + 8iT - 37T^{2} \) |
| 41 | \( 1 - 12T + 41T^{2} \) |
| 43 | \( 1 + 8iT - 43T^{2} \) |
| 47 | \( 1 - 4iT - 47T^{2} \) |
| 53 | \( 1 + 4iT - 53T^{2} \) |
| 59 | \( 1 - 8T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 - 2iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 8iT - 97T^{2} \) |
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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
−8.892504296088618306004132580530, −7.55236634759414814958850890845, −7.32171732666136781469399155876, −6.71286856195478886250858434062, −5.82945191127033184202905508434, −4.05801419980960362158142521448, −3.83462340453073112989992800959, −2.58569693804047336298394069932, −1.72737423349988690003199800997, −0.31822386352142298968216031437,
1.90800047587759104968312137035, 2.83881055560240536008856200296, 4.14916561742667470191205768069, 5.01660817728515259090248791062, 5.80353173378479176327032114960, 6.23735372577332937544338302181, 7.48287639462489951606624654433, 8.392466054178223268517297026847, 8.620159437641553701178842379971, 9.406611522312070419945835214748