| L(s) = 1 | + (0.527 + 0.235i)2-s + (−1.46 − 0.924i)3-s + (−1.11 − 1.23i)4-s + (2.74 − 1.22i)5-s + (−0.555 − 0.832i)6-s + (1.13 − 0.242i)7-s + (−0.654 − 2.01i)8-s + (1.28 + 2.70i)9-s + 1.73·10-s + (−2.54 − 2.12i)11-s + (0.487 + 2.84i)12-s + (0.412 + 3.92i)13-s + (0.658 + 0.139i)14-s + (−5.14 − 0.749i)15-s + (−0.220 + 2.09i)16-s + (0.254 + 0.184i)17-s + ⋯ |
| L(s) = 1 | + (0.373 + 0.166i)2-s + (−0.845 − 0.534i)3-s + (−0.557 − 0.619i)4-s + (1.22 − 0.546i)5-s + (−0.226 − 0.339i)6-s + (0.430 − 0.0914i)7-s + (−0.231 − 0.712i)8-s + (0.429 + 0.902i)9-s + 0.548·10-s + (−0.768 − 0.640i)11-s + (0.140 + 0.821i)12-s + (0.114 + 1.08i)13-s + (0.175 + 0.0373i)14-s + (−1.32 − 0.193i)15-s + (−0.0550 + 0.524i)16-s + (0.0616 + 0.0447i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.591 + 0.806i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.591 + 0.806i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.892143 - 0.451981i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.892143 - 0.451981i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.46 + 0.924i)T \) |
| 11 | \( 1 + (2.54 + 2.12i)T \) |
| good | 2 | \( 1 + (-0.527 - 0.235i)T + (1.33 + 1.48i)T^{2} \) |
| 5 | \( 1 + (-2.74 + 1.22i)T + (3.34 - 3.71i)T^{2} \) |
| 7 | \( 1 + (-1.13 + 0.242i)T + (6.39 - 2.84i)T^{2} \) |
| 13 | \( 1 + (-0.412 - 3.92i)T + (-12.7 + 2.70i)T^{2} \) |
| 17 | \( 1 + (-0.254 - 0.184i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-1.96 - 6.05i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (-0.0427 + 0.0740i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-7.53 + 1.60i)T + (26.4 - 11.7i)T^{2} \) |
| 31 | \( 1 + (0.682 + 6.49i)T + (-30.3 + 6.44i)T^{2} \) |
| 37 | \( 1 + (1.92 - 5.93i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (5.68 + 1.20i)T + (37.4 + 16.6i)T^{2} \) |
| 43 | \( 1 + (-3.39 - 5.87i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (0.219 - 0.243i)T + (-4.91 - 46.7i)T^{2} \) |
| 53 | \( 1 + (-1.95 + 1.42i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (1.53 + 1.70i)T + (-6.16 + 58.6i)T^{2} \) |
| 61 | \( 1 + (-1.43 + 13.6i)T + (-59.6 - 12.6i)T^{2} \) |
| 67 | \( 1 + (5.83 - 10.1i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (7.05 + 5.12i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-0.910 + 2.80i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (8.11 + 3.61i)T + (52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (0.518 - 4.93i)T + (-81.1 - 17.2i)T^{2} \) |
| 89 | \( 1 - 2.12T + 89T^{2} \) |
| 97 | \( 1 + (0.0811 + 0.0361i)T + (64.9 + 72.0i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.67971345824154963094925393400, −12.99166259395010027567650986644, −11.78632688557435817364510974709, −10.45908251901021112074369669188, −9.621626997339676296601786724505, −8.190373134127764145045467126105, −6.38298733504434709723108420122, −5.63717800458879999034524388817, −4.67197323752618179810404117426, −1.52326334030233110988760299592,
2.90566720181488017870896500950, 4.81961264223146835033459627570, 5.56469015100982285540247434811, 7.12883917830682621950810445107, 8.791245310435368547849096061588, 10.02946195764326785152818308106, 10.76383368123247501689842599767, 12.06112806753231757433676861647, 13.00158721674515077643251697346, 13.89255316822799035584782519639
