| L(s) = 1 | + (−1.52 + 0.944i)2-s + (−0.431 − 1.51i)3-s + (0.542 − 1.08i)4-s + (−0.593 + 0.229i)5-s + (2.08 + 1.90i)6-s + (0.343 − 3.70i)7-s + (−0.129 − 1.39i)8-s + (0.437 − 0.270i)9-s + (0.687 − 0.910i)10-s + (0.539 − 0.333i)11-s + (−1.88 − 0.352i)12-s + (0.285 − 3.08i)13-s + (2.97 + 5.97i)14-s + (0.604 + 0.800i)15-s + (2.98 + 3.95i)16-s + (−3.28 + 2.99i)17-s + ⋯ |
| L(s) = 1 | + (−1.07 + 0.667i)2-s + (−0.249 − 0.875i)3-s + (0.271 − 0.544i)4-s + (−0.265 + 0.102i)5-s + (0.853 + 0.777i)6-s + (0.129 − 1.39i)7-s + (−0.0458 − 0.494i)8-s + (0.145 − 0.0902i)9-s + (0.217 − 0.287i)10-s + (0.162 − 0.100i)11-s + (−0.544 − 0.101i)12-s + (0.0793 − 0.855i)13-s + (0.794 + 1.59i)14-s + (0.156 + 0.206i)15-s + (0.746 + 0.988i)16-s + (−0.796 + 0.726i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.520 + 0.853i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.520 + 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.440739 - 0.247535i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.440739 - 0.247535i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 103 | \( 1 + (2.10 - 9.92i)T \) |
| good | 2 | \( 1 + (1.52 - 0.944i)T + (0.891 - 1.79i)T^{2} \) |
| 3 | \( 1 + (0.431 + 1.51i)T + (-2.55 + 1.57i)T^{2} \) |
| 5 | \( 1 + (0.593 - 0.229i)T + (3.69 - 3.36i)T^{2} \) |
| 7 | \( 1 + (-0.343 + 3.70i)T + (-6.88 - 1.28i)T^{2} \) |
| 11 | \( 1 + (-0.539 + 0.333i)T + (4.90 - 9.84i)T^{2} \) |
| 13 | \( 1 + (-0.285 + 3.08i)T + (-12.7 - 2.38i)T^{2} \) |
| 17 | \( 1 + (3.28 - 2.99i)T + (1.56 - 16.9i)T^{2} \) |
| 19 | \( 1 + (-0.968 + 3.40i)T + (-16.1 - 10.0i)T^{2} \) |
| 23 | \( 1 + (-2.32 - 1.43i)T + (10.2 + 20.5i)T^{2} \) |
| 29 | \( 1 + (3.69 - 1.43i)T + (21.4 - 19.5i)T^{2} \) |
| 31 | \( 1 + (-0.661 - 0.875i)T + (-8.48 + 29.8i)T^{2} \) |
| 37 | \( 1 + (-6.60 - 1.23i)T + (34.5 + 13.3i)T^{2} \) |
| 41 | \( 1 + (-8.56 - 3.31i)T + (30.2 + 27.6i)T^{2} \) |
| 43 | \( 1 + (-9.59 + 1.79i)T + (40.0 - 15.5i)T^{2} \) |
| 47 | \( 1 - 1.07T + 47T^{2} \) |
| 53 | \( 1 + (-3.21 + 11.3i)T + (-45.0 - 27.9i)T^{2} \) |
| 59 | \( 1 + (-0.370 - 4.00i)T + (-57.9 + 10.8i)T^{2} \) |
| 61 | \( 1 + (8.51 - 7.76i)T + (5.62 - 60.7i)T^{2} \) |
| 67 | \( 1 + (-0.738 - 7.96i)T + (-65.8 + 12.3i)T^{2} \) |
| 71 | \( 1 + (-1.77 - 0.688i)T + (52.4 + 47.8i)T^{2} \) |
| 73 | \( 1 + (8.52 + 3.30i)T + (53.9 + 49.1i)T^{2} \) |
| 79 | \( 1 + (-15.4 + 5.99i)T + (58.3 - 53.2i)T^{2} \) |
| 83 | \( 1 + (0.0694 - 0.749i)T + (-81.5 - 15.2i)T^{2} \) |
| 89 | \( 1 + (3.95 + 7.93i)T + (-53.6 + 71.0i)T^{2} \) |
| 97 | \( 1 + (-8.56 - 7.81i)T + (8.95 + 96.5i)T^{2} \) |
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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
−13.35130720818850146944871608180, −12.90362326916374962792256489007, −11.30665915645068397846652289772, −10.33328544391277392503268625122, −9.111658077240909247933690645940, −7.68968354008314724576717230712, −7.32648843649932392661028338291, −6.22017027522478749868956346590, −3.97293525575344694911678076735, −0.911464866365109881968580062826,
2.28067257358111972021643653229, 4.43295389099027833836046782702, 5.79243609194039029335565349158, 7.79010932397175130723347747111, 9.178616722853477373284429802221, 9.409711637264019769386247540889, 10.79328249511250011964942598449, 11.51931419631041071312899738987, 12.41600977490118167337569109795, 14.15053103474624188695321967720
