L(s) = 1 | − 0.271i·2-s − 0.319·3-s + 1.92·4-s + i·5-s + 0.0867i·6-s + 3.38i·7-s − 1.06i·8-s − 2.89·9-s + 0.271·10-s + 1.75i·11-s − 0.615·12-s + 0.917·14-s − 0.319i·15-s + 3.56·16-s − 1.95·17-s + 0.786i·18-s + ⋯ |
L(s) = 1 | − 0.191i·2-s − 0.184·3-s + 0.963·4-s + 0.447i·5-s + 0.0354i·6-s + 1.27i·7-s − 0.376i·8-s − 0.965·9-s + 0.0858·10-s + 0.528i·11-s − 0.177·12-s + 0.245·14-s − 0.0825i·15-s + 0.890·16-s − 0.473·17-s + 0.185i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0304 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0304 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.02779 + 0.996929i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.02779 + 0.996929i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - iT \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + 0.271iT - 2T^{2} \) |
| 3 | \( 1 + 0.319T + 3T^{2} \) |
| 7 | \( 1 - 3.38iT - 7T^{2} \) |
| 11 | \( 1 - 1.75iT - 11T^{2} \) |
| 17 | \( 1 + 1.95T + 17T^{2} \) |
| 19 | \( 1 - 7.13iT - 19T^{2} \) |
| 23 | \( 1 + 7.61T + 23T^{2} \) |
| 29 | \( 1 - 3.98T + 29T^{2} \) |
| 31 | \( 1 + 4.86iT - 31T^{2} \) |
| 37 | \( 1 - 10.5iT - 37T^{2} \) |
| 41 | \( 1 + 0.911iT - 41T^{2} \) |
| 43 | \( 1 + 4.58T + 43T^{2} \) |
| 47 | \( 1 + 8.58iT - 47T^{2} \) |
| 53 | \( 1 - 11.9T + 53T^{2} \) |
| 59 | \( 1 - 3.82iT - 59T^{2} \) |
| 61 | \( 1 - 7.98T + 61T^{2} \) |
| 67 | \( 1 + 0.472iT - 67T^{2} \) |
| 71 | \( 1 - 5.50iT - 71T^{2} \) |
| 73 | \( 1 - 2.93iT - 73T^{2} \) |
| 79 | \( 1 - 4.09T + 79T^{2} \) |
| 83 | \( 1 - 11.7iT - 83T^{2} \) |
| 89 | \( 1 + 3.85iT - 89T^{2} \) |
| 97 | \( 1 - 6.95iT - 97T^{2} \) |
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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
−10.31605512135751073087347608012, −9.874945048164385088060088090141, −8.528366694973383923483447961150, −7.995379266787846482664459194049, −6.75336819460804864644737767705, −6.04386623381263465127563915434, −5.43720000321315204351255894225, −3.81915299416774342564947679393, −2.66090431641148807609742710695, −1.97021588517625936823118546930,
0.66157579030691434203374443346, 2.30369687894783428480446189376, 3.48711040799218681781546384016, 4.66726372379214083974466866960, 5.73852755824858119877787659591, 6.55973782043030958126318447781, 7.33280047133422594812378342386, 8.210793213438598996405457416603, 9.003070188892429008332774029525, 10.24045938291718214024946077469