L(s) = 1 | + (0.707 − 0.707i)2-s − 1.00i·4-s + (0.707 + 0.707i)7-s + (−0.707 − 0.707i)8-s − 0.585i·11-s + (3.43 − 3.43i)13-s + 1.00·14-s − 1.00·16-s + (−0.906 + 0.906i)17-s − 2.04i·19-s + (−0.414 − 0.414i)22-s + (0.257 + 0.257i)23-s − 4.86i·26-s + (0.707 − 0.707i)28-s + 1.75·29-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s − 0.500i·4-s + (0.267 + 0.267i)7-s + (−0.250 − 0.250i)8-s − 0.176i·11-s + (0.953 − 0.953i)13-s + 0.267·14-s − 0.250·16-s + (−0.219 + 0.219i)17-s − 0.470i·19-s + (−0.0883 − 0.0883i)22-s + (0.0537 + 0.0537i)23-s − 0.953i·26-s + (0.133 − 0.133i)28-s + 0.325·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.161 + 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.161 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.420489388\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.420489388\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 0.707i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.707 - 0.707i)T \) |
good | 11 | \( 1 + 0.585iT - 11T^{2} \) |
| 13 | \( 1 + (-3.43 + 3.43i)T - 13iT^{2} \) |
| 17 | \( 1 + (0.906 - 0.906i)T - 17iT^{2} \) |
| 19 | \( 1 + 2.04iT - 19T^{2} \) |
| 23 | \( 1 + (-0.257 - 0.257i)T + 23iT^{2} \) |
| 29 | \( 1 - 1.75T + 29T^{2} \) |
| 31 | \( 1 - 1.47T + 31T^{2} \) |
| 37 | \( 1 + (4.04 + 4.04i)T + 37iT^{2} \) |
| 41 | \( 1 + 3.84iT - 41T^{2} \) |
| 43 | \( 1 + (-5.10 + 5.10i)T - 43iT^{2} \) |
| 47 | \( 1 + (5.49 - 5.49i)T - 47iT^{2} \) |
| 53 | \( 1 + (-5.02 - 5.02i)T + 53iT^{2} \) |
| 59 | \( 1 - 11.1T + 59T^{2} \) |
| 61 | \( 1 - 3.78T + 61T^{2} \) |
| 67 | \( 1 + (6.74 + 6.74i)T + 67iT^{2} \) |
| 71 | \( 1 + 10.9iT - 71T^{2} \) |
| 73 | \( 1 + (-5.97 + 5.97i)T - 73iT^{2} \) |
| 79 | \( 1 + 0.944iT - 79T^{2} \) |
| 83 | \( 1 + (7.82 + 7.82i)T + 83iT^{2} \) |
| 89 | \( 1 - 0.0705T + 89T^{2} \) |
| 97 | \( 1 + (0.746 + 0.746i)T + 97iT^{2} \) |
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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
−8.601574829737921551433887601150, −7.79405665097103654765337924836, −6.86009629231643252765912920839, −5.97234057848533496929677316771, −5.43854088234923764783426480171, −4.53651022248332944240976170123, −3.66106442917293034802342600058, −2.88431323407144779324008155088, −1.86765048066114449724682218052, −0.67923771605397600615623007670,
1.26838356161347223521965883619, 2.43807747633239908952576530615, 3.60597550659190002589047977099, 4.23832554859170918579580674809, 5.04108036976049561386903246958, 5.89237163461373762614315677441, 6.70468539584213248320630168336, 7.14674657267436146765934897770, 8.277105233993912606965606223190, 8.542051834473055149171953538816