L(s) = 1 | + (1 − 2i)5-s − 3·9-s + 6·13-s − 8i·17-s + (−3 − 4i)25-s − 4i·29-s + 2·37-s − 10·41-s + (−3 + 6i)45-s + 7·49-s + 14·53-s − 12i·61-s + (6 − 12i)65-s + 16i·73-s + 9·81-s + ⋯ |
L(s) = 1 | + (0.447 − 0.894i)5-s − 9-s + 1.66·13-s − 1.94i·17-s + (−0.600 − 0.800i)25-s − 0.742i·29-s + 0.328·37-s − 1.56·41-s + (−0.447 + 0.894i)45-s + 49-s + 1.92·53-s − 1.53i·61-s + (0.744 − 1.48i)65-s + 1.87i·73-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.316 + 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.316 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.19616 - 0.862150i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.19616 - 0.862150i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-1 + 2i)T \) |
good | 3 | \( 1 + 3T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 6T + 13T^{2} \) |
| 17 | \( 1 + 8iT - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 4iT - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 14T + 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 + 12iT - 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 16iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 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
−10.35795920313683108427023191836, −9.337043635161965010128317803859, −8.750650823645753158718143540585, −8.014066743832214300646375314543, −6.70345157950584420237967460650, −5.73681472321274272716718651273, −5.05080280077562764934099287219, −3.76762221979729374103780599959, −2.46820928880916121490131803546, −0.846342616596343495164064277755,
1.70994289517454519158336061999, 3.10390762895250756848351725549, 3.93973282623656955346345631119, 5.68699361426989069092165727334, 6.08817597828249896892998079615, 7.07199505435686435711251716096, 8.395248025325154176219627201522, 8.760262332511950030321180966448, 10.12808288868102235352532880412, 10.75303934526943902420996081597