| L(s) = 1 | + (−0.0151 + 0.0286i)2-s + (−2.04 + 1.93i)3-s + (1.12 + 1.65i)4-s + (−1.63 + 0.360i)5-s + (−0.0244 − 0.0879i)6-s + (3.31 − 2.51i)7-s + (−0.128 + 0.0140i)8-s + (0.266 − 4.91i)9-s + (0.0145 − 0.0524i)10-s + (1.56 + 3.92i)11-s + (−5.49 − 1.20i)12-s + (−0.216 − 3.99i)13-s + (0.0218 + 0.133i)14-s + (2.65 − 3.90i)15-s + (−1.47 + 3.71i)16-s + (−1.05 − 0.804i)17-s + ⋯ |
| L(s) = 1 | + (−0.0107 + 0.0202i)2-s + (−1.17 + 1.11i)3-s + (0.560 + 0.827i)4-s + (−0.732 + 0.161i)5-s + (−0.00997 − 0.0359i)6-s + (1.25 − 0.952i)7-s + (−0.0455 + 0.00495i)8-s + (0.0888 − 1.63i)9-s + (0.00460 − 0.0165i)10-s + (0.471 + 1.18i)11-s + (−1.58 − 0.349i)12-s + (−0.0600 − 1.10i)13-s + (0.00583 + 0.0355i)14-s + (0.684 − 1.00i)15-s + (−0.369 + 0.927i)16-s + (−0.256 − 0.195i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.226 - 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.226 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.547284 + 0.434819i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.547284 + 0.434819i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 59 | \( 1 + (-7.09 + 2.94i)T \) |
| good | 2 | \( 1 + (0.0151 - 0.0286i)T + (-1.12 - 1.65i)T^{2} \) |
| 3 | \( 1 + (2.04 - 1.93i)T + (0.162 - 2.99i)T^{2} \) |
| 5 | \( 1 + (1.63 - 0.360i)T + (4.53 - 2.09i)T^{2} \) |
| 7 | \( 1 + (-3.31 + 2.51i)T + (1.87 - 6.74i)T^{2} \) |
| 11 | \( 1 + (-1.56 - 3.92i)T + (-7.98 + 7.56i)T^{2} \) |
| 13 | \( 1 + (0.216 + 3.99i)T + (-12.9 + 1.40i)T^{2} \) |
| 17 | \( 1 + (1.05 + 0.804i)T + (4.54 + 16.3i)T^{2} \) |
| 19 | \( 1 + (-1.15 + 0.390i)T + (15.1 - 11.4i)T^{2} \) |
| 23 | \( 1 + (-5.06 + 3.04i)T + (10.7 - 20.3i)T^{2} \) |
| 29 | \( 1 + (1.98 + 3.73i)T + (-16.2 + 24.0i)T^{2} \) |
| 31 | \( 1 + (1.43 + 0.481i)T + (24.6 + 18.7i)T^{2} \) |
| 37 | \( 1 + (-0.590 - 0.0642i)T + (36.1 + 7.95i)T^{2} \) |
| 41 | \( 1 + (0.645 + 0.388i)T + (19.2 + 36.2i)T^{2} \) |
| 43 | \( 1 + (2.49 - 6.25i)T + (-31.2 - 29.5i)T^{2} \) |
| 47 | \( 1 + (6.15 + 1.35i)T + (42.6 + 19.7i)T^{2} \) |
| 53 | \( 1 + (3.01 + 10.8i)T + (-45.4 + 27.3i)T^{2} \) |
| 61 | \( 1 + (0.100 - 0.188i)T + (-34.2 - 50.4i)T^{2} \) |
| 67 | \( 1 + (1.91 - 0.208i)T + (65.4 - 14.4i)T^{2} \) |
| 71 | \( 1 + (-5.91 - 1.30i)T + (64.4 + 29.8i)T^{2} \) |
| 73 | \( 1 + (-1.75 - 10.7i)T + (-69.1 + 23.3i)T^{2} \) |
| 79 | \( 1 + (0.204 + 0.193i)T + (4.27 + 78.8i)T^{2} \) |
| 83 | \( 1 + (10.3 - 12.2i)T + (-13.4 - 81.9i)T^{2} \) |
| 89 | \( 1 + (3.82 + 7.22i)T + (-49.9 + 73.6i)T^{2} \) |
| 97 | \( 1 + (0.626 - 3.82i)T + (-91.9 - 30.9i)T^{2} \) |
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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
−15.46434209003905958067311398930, −14.82563648651310702448371890377, −12.78719428442109081511759588602, −11.49027271073160003382499940501, −11.19216798580854950839017132441, −10.00059979757113482583966297052, −8.018751781877043507747486524190, −6.95838250104288709880783083332, −4.92463660442825794007184684053, −3.91526561078440413733712276410,
1.56645888086123593298427355711, 5.13561363028867290882702483687, 6.15880889909691617281424557680, 7.37724175715164310681883363609, 8.825581381438202320405906210159, 11.13837371645511936171312522242, 11.42786674262584935197333812535, 12.17238195577225237180294623683, 13.81745529165607068492545776090, 14.93504194807620913333601965887
