| L(s) = 1 | + (0.407 − 0.540i)2-s + (−1.35 − 2.72i)3-s + (0.421 + 1.48i)4-s + (−0.246 − 2.65i)5-s + (−2.02 − 0.378i)6-s + (−2.18 + 0.845i)7-s + (2.23 + 0.866i)8-s + (−3.76 + 4.98i)9-s + (−1.53 − 0.950i)10-s + (2.06 − 2.73i)11-s + (3.46 − 3.15i)12-s + (6.56 − 2.54i)13-s + (−0.433 + 1.52i)14-s + (−6.89 + 4.26i)15-s + (−1.24 + 0.768i)16-s + (1.40 − 0.261i)17-s + ⋯ |
| L(s) = 1 | + (0.288 − 0.381i)2-s + (−0.782 − 1.57i)3-s + (0.210 + 0.741i)4-s + (−0.110 − 1.18i)5-s + (−0.826 − 0.154i)6-s + (−0.824 + 0.319i)7-s + (0.790 + 0.306i)8-s + (−1.25 + 1.66i)9-s + (−0.485 − 0.300i)10-s + (0.623 − 0.825i)11-s + (1.00 − 0.911i)12-s + (1.82 − 0.705i)13-s + (−0.115 + 0.407i)14-s + (−1.78 + 1.10i)15-s + (−0.310 + 0.192i)16-s + (0.339 − 0.0634i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.329 + 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.329 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.548750 - 0.772671i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.548750 - 0.772671i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 103 | \( 1 + (9.85 + 2.42i)T \) |
| good | 2 | \( 1 + (-0.407 + 0.540i)T + (-0.547 - 1.92i)T^{2} \) |
| 3 | \( 1 + (1.35 + 2.72i)T + (-1.80 + 2.39i)T^{2} \) |
| 5 | \( 1 + (0.246 + 2.65i)T + (-4.91 + 0.918i)T^{2} \) |
| 7 | \( 1 + (2.18 - 0.845i)T + (5.17 - 4.71i)T^{2} \) |
| 11 | \( 1 + (-2.06 + 2.73i)T + (-3.01 - 10.5i)T^{2} \) |
| 13 | \( 1 + (-6.56 + 2.54i)T + (9.60 - 8.75i)T^{2} \) |
| 17 | \( 1 + (-1.40 + 0.261i)T + (15.8 - 6.14i)T^{2} \) |
| 19 | \( 1 + (2.98 - 6.00i)T + (-11.4 - 15.1i)T^{2} \) |
| 23 | \( 1 + (-1.36 - 1.80i)T + (-6.29 + 22.1i)T^{2} \) |
| 29 | \( 1 + (-0.140 - 1.51i)T + (-28.5 + 5.32i)T^{2} \) |
| 31 | \( 1 + (-0.201 + 0.124i)T + (13.8 - 27.7i)T^{2} \) |
| 37 | \( 1 + (0.281 - 0.256i)T + (3.41 - 36.8i)T^{2} \) |
| 41 | \( 1 + (-0.163 + 1.76i)T + (-40.3 - 7.53i)T^{2} \) |
| 43 | \( 1 + (6.49 + 5.92i)T + (3.96 + 42.8i)T^{2} \) |
| 47 | \( 1 + 3.45T + 47T^{2} \) |
| 53 | \( 1 + (2.91 - 5.86i)T + (-31.9 - 42.2i)T^{2} \) |
| 59 | \( 1 + (1.48 + 0.575i)T + (43.6 + 39.7i)T^{2} \) |
| 61 | \( 1 + (-2.56 + 0.479i)T + (56.8 - 22.0i)T^{2} \) |
| 67 | \( 1 + (-2.16 - 0.839i)T + (49.5 + 45.1i)T^{2} \) |
| 71 | \( 1 + (0.352 - 3.80i)T + (-69.7 - 13.0i)T^{2} \) |
| 73 | \( 1 + (0.469 - 5.06i)T + (-71.7 - 13.4i)T^{2} \) |
| 79 | \( 1 + (-0.693 - 7.48i)T + (-77.6 + 14.5i)T^{2} \) |
| 83 | \( 1 + (-14.8 + 5.76i)T + (61.3 - 55.9i)T^{2} \) |
| 89 | \( 1 + (0.654 - 2.30i)T + (-75.6 - 46.8i)T^{2} \) |
| 97 | \( 1 + (-7.33 - 1.37i)T + (90.4 + 35.0i)T^{2} \) |
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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
−13.06819848068844505786062777982, −12.54701385065678756159211796248, −11.83923936799411726973466378279, −10.86531584767631105806240101091, −8.660338368403119904090226771283, −8.044434118841144313513205633243, −6.51727936749974786195648762355, −5.63737999412423686992148459542, −3.53408413667716147289434481691, −1.33908257298734438354506633320,
3.59635611579018542886920302926, 4.71023762954090996930283478137, 6.38833925821128387990130119668, 6.60626353094023200089491242250, 9.209222800402326736344897806109, 10.09758066627906474418450499871, 10.87977717410882421155204885736, 11.44290929120718870382995265549, 13.38843068554006865372738762599, 14.62212552578133161875521754178
