L(s) = 1 | − 2i·5-s + 2·7-s + 3·9-s + 2i·11-s − i·13-s + 6·17-s − 6i·19-s − 8·23-s + 25-s + 2i·29-s + 10·31-s − 4i·35-s + 6i·37-s + 6·41-s − 4i·43-s + ⋯ |
L(s) = 1 | − 0.894i·5-s + 0.755·7-s + 9-s + 0.603i·11-s − 0.277i·13-s + 1.45·17-s − 1.37i·19-s − 1.66·23-s + 0.200·25-s + 0.371i·29-s + 1.79·31-s − 0.676i·35-s + 0.986i·37-s + 0.937·41-s − 0.609i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.391387640\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.391387640\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 13 | \( 1 + iT \) |
good | 3 | \( 1 - 3T^{2} \) |
| 5 | \( 1 + 2iT - 5T^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 - 2iT - 11T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 + 6iT - 19T^{2} \) |
| 23 | \( 1 + 8T + 23T^{2} \) |
| 29 | \( 1 - 2iT - 29T^{2} \) |
| 31 | \( 1 - 10T + 31T^{2} \) |
| 37 | \( 1 - 6iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 + 2T + 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 - 10iT - 59T^{2} \) |
| 61 | \( 1 + 2iT - 61T^{2} \) |
| 67 | \( 1 - 10iT - 67T^{2} \) |
| 71 | \( 1 + 10T + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−8.435339999531146319969007556001, −7.85052329033276788798317621047, −7.20964060717829026776974784389, −6.27567747924984960660861854308, −5.27643298662796075001281096075, −4.68164800387422122711270661767, −4.13920696068713736577114544821, −2.85609335182729288188206440351, −1.67257766671392312969540112048, −0.894354999349218799491935799062,
1.14906112456943752191491702303, 2.10261292904118788852418234774, 3.24384143830868426317446865170, 3.99040548654153492098683079604, 4.80514747541410603419948792692, 5.97648036585378566945259649141, 6.27766473415904451839885481166, 7.51739694025035837376805256081, 7.77674216922836956532674511703, 8.511218615322201476125324886202