| L(s) = 1 | + (0.283 + 0.0529i)2-s + (0.135 − 1.46i)3-s + (−1.78 − 0.692i)4-s + (1.97 − 2.61i)5-s + (0.115 − 0.406i)6-s + (−3.42 + 2.12i)7-s + (−0.959 − 0.594i)8-s + (0.831 + 0.155i)9-s + (0.697 − 0.635i)10-s + (5.08 + 0.950i)11-s + (−1.25 + 2.51i)12-s + (1.07 − 0.663i)13-s + (−1.08 + 0.419i)14-s + (−3.55 − 3.23i)15-s + (2.59 + 2.36i)16-s + (0.813 + 2.86i)17-s + ⋯ |
| L(s) = 1 | + (0.200 + 0.0374i)2-s + (0.0781 − 0.843i)3-s + (−0.893 − 0.346i)4-s + (0.883 − 1.16i)5-s + (0.0472 − 0.166i)6-s + (−1.29 + 0.802i)7-s + (−0.339 − 0.210i)8-s + (0.277 + 0.0518i)9-s + (0.220 − 0.201i)10-s + (1.53 + 0.286i)11-s + (−0.362 + 0.727i)12-s + (0.297 − 0.183i)13-s + (−0.289 + 0.112i)14-s + (−0.917 − 0.836i)15-s + (0.648 + 0.590i)16-s + (0.197 + 0.693i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.353 + 0.935i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.353 + 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.857355 - 0.592287i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.857355 - 0.592287i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 103 | \( 1 + (5.29 + 8.66i)T \) |
| good | 2 | \( 1 + (-0.283 - 0.0529i)T + (1.86 + 0.722i)T^{2} \) |
| 3 | \( 1 + (-0.135 + 1.46i)T + (-2.94 - 0.551i)T^{2} \) |
| 5 | \( 1 + (-1.97 + 2.61i)T + (-1.36 - 4.80i)T^{2} \) |
| 7 | \( 1 + (3.42 - 2.12i)T + (3.12 - 6.26i)T^{2} \) |
| 11 | \( 1 + (-5.08 - 0.950i)T + (10.2 + 3.97i)T^{2} \) |
| 13 | \( 1 + (-1.07 + 0.663i)T + (5.79 - 11.6i)T^{2} \) |
| 17 | \( 1 + (-0.813 - 2.86i)T + (-14.4 + 8.94i)T^{2} \) |
| 19 | \( 1 + (-0.0430 - 0.464i)T + (-18.6 + 3.49i)T^{2} \) |
| 23 | \( 1 + (7.91 - 1.47i)T + (21.4 - 8.30i)T^{2} \) |
| 29 | \( 1 + (0.451 - 0.597i)T + (-7.93 - 27.8i)T^{2} \) |
| 31 | \( 1 + (0.774 + 0.706i)T + (2.86 + 30.8i)T^{2} \) |
| 37 | \( 1 + (0.0972 - 0.195i)T + (-22.2 - 29.5i)T^{2} \) |
| 41 | \( 1 + (3.15 + 4.17i)T + (-11.2 + 39.4i)T^{2} \) |
| 43 | \( 1 + (-4.62 - 9.29i)T + (-25.9 + 34.3i)T^{2} \) |
| 47 | \( 1 + 1.28T + 47T^{2} \) |
| 53 | \( 1 + (-0.387 - 4.18i)T + (-52.0 + 9.73i)T^{2} \) |
| 59 | \( 1 + (-4.13 - 2.55i)T + (26.2 + 52.8i)T^{2} \) |
| 61 | \( 1 + (3.30 + 11.6i)T + (-51.8 + 32.1i)T^{2} \) |
| 67 | \( 1 + (12.2 + 7.60i)T + (29.8 + 59.9i)T^{2} \) |
| 71 | \( 1 + (-4.61 - 6.10i)T + (-19.4 + 68.2i)T^{2} \) |
| 73 | \( 1 + (-6.22 - 8.24i)T + (-19.9 + 70.2i)T^{2} \) |
| 79 | \( 1 + (9.31 - 12.3i)T + (-21.6 - 75.9i)T^{2} \) |
| 83 | \( 1 + (-4.51 + 2.79i)T + (36.9 - 74.2i)T^{2} \) |
| 89 | \( 1 + (12.5 - 4.86i)T + (65.7 - 59.9i)T^{2} \) |
| 97 | \( 1 + (0.454 - 1.59i)T + (-82.4 - 51.0i)T^{2} \) |
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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
−13.43932822369512202248135635651, −12.60891616517472521400815596963, −12.27958476643889084368068456420, −9.845769615835589703957295976424, −9.383795696760080749879580670891, −8.378740314950748589360036466133, −6.40403953484165345279453429235, −5.74383315200491288933595083749, −4.06827245701206022309221109372, −1.51912774190728227925667876780,
3.35346334398239633431032713465, 4.12583093628447769030875662043, 6.07103270321774742851572628726, 7.04790425021014189248117293426, 9.081537725716015308109954135527, 9.798185731207851402032066014702, 10.40786584749998075916320391467, 11.96665363009743877546082538681, 13.34968248049502279524745652412, 13.98971150343107759489762885556
