| L(s) = 1 | + (−0.253 + 1.44i)2-s + (1.15 + 0.970i)3-s + (−0.130 − 0.0473i)4-s + (0.939 − 0.342i)5-s + (−1.69 + 1.41i)6-s + (−2.03 − 3.52i)7-s + (−1.36 + 2.35i)8-s + (−0.124 − 0.707i)9-s + (0.253 + 1.44i)10-s + (0.310 − 0.537i)11-s + (−0.104 − 0.180i)12-s + (−3.90 + 3.27i)13-s + (5.59 − 2.03i)14-s + (1.41 + 0.516i)15-s + (−3.26 − 2.73i)16-s + (−0.0462 + 0.262i)17-s + ⋯ |
| L(s) = 1 | + (−0.179 + 1.01i)2-s + (0.667 + 0.560i)3-s + (−0.0650 − 0.0236i)4-s + (0.420 − 0.152i)5-s + (−0.690 + 0.579i)6-s + (−0.769 − 1.33i)7-s + (−0.481 + 0.833i)8-s + (−0.0416 − 0.235i)9-s + (0.0802 + 0.455i)10-s + (0.0936 − 0.162i)11-s + (−0.0301 − 0.0522i)12-s + (−1.08 + 0.908i)13-s + (1.49 − 0.544i)14-s + (0.366 + 0.133i)15-s + (−0.815 − 0.684i)16-s + (−0.0112 + 0.0636i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.107 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.107 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.839897 + 0.753989i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.839897 + 0.753989i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-0.939 + 0.342i)T \) |
| 19 | \( 1 + (-0.399 + 4.34i)T \) |
| good | 2 | \( 1 + (0.253 - 1.44i)T + (-1.87 - 0.684i)T^{2} \) |
| 3 | \( 1 + (-1.15 - 0.970i)T + (0.520 + 2.95i)T^{2} \) |
| 7 | \( 1 + (2.03 + 3.52i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.310 + 0.537i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (3.90 - 3.27i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (0.0462 - 0.262i)T + (-15.9 - 5.81i)T^{2} \) |
| 23 | \( 1 + (-5.48 - 1.99i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-0.708 - 4.01i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (-3.24 - 5.62i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 8.83T + 37T^{2} \) |
| 41 | \( 1 + (3.43 + 2.88i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-1.69 + 0.615i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-2.00 - 11.3i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (-2.37 - 0.862i)T + (40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (-0.154 + 0.876i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (2.03 + 0.742i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (2.44 + 13.8i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (-1.54 + 0.563i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (-1.37 - 1.15i)T + (12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (-3.94 - 3.30i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (6.89 + 11.9i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (0.000572 - 0.000480i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (1.80 - 10.2i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−14.30527965386350990133060413364, −13.73320957889230957647674984210, −12.26482280726555867152288573078, −10.75361858613130072407571142879, −9.533750526077324939625938369846, −8.829958145076540217456893086264, −7.23121084155084848469108975705, −6.62991386472396369970432759585, −4.80691716312545596027625271473, −3.13322828693238067869541436491,
2.21864585415561968364787383970, 3.00560569533556839305915292474, 5.58704825034814085234511427536, 6.96556783372984767634461820413, 8.463793573261465772462967067207, 9.595144331998264863537544906964, 10.37453645319689038846648832744, 11.87012209881355837904827208822, 12.57836150034295642750856361332, 13.35722372486157772349348810588
