| L(s) = 1 | + (−0.483 − 2.11i)2-s + (−0.240 − 0.222i)3-s + (−2.45 + 1.18i)4-s + (0.0260 − 0.00392i)5-s + (−0.356 + 0.616i)6-s + (1.56 + 2.71i)7-s + (0.984 + 1.23i)8-s + (−0.216 − 2.88i)9-s + (−0.0208 − 0.0532i)10-s + (3.36 + 1.62i)11-s + (0.853 + 0.263i)12-s + (−1.78 + 4.55i)13-s + (4.98 − 4.62i)14-s + (−0.00712 − 0.00485i)15-s + (−1.25 + 1.57i)16-s + (−5.84 − 0.881i)17-s + ⋯ |
| L(s) = 1 | + (−0.342 − 1.49i)2-s + (−0.138 − 0.128i)3-s + (−1.22 + 0.591i)4-s + (0.0116 − 0.00175i)5-s + (−0.145 + 0.251i)6-s + (0.591 + 1.02i)7-s + (0.348 + 0.436i)8-s + (−0.0720 − 0.961i)9-s + (−0.00660 − 0.0168i)10-s + (1.01 + 0.488i)11-s + (0.246 + 0.0760i)12-s + (−0.496 + 1.26i)13-s + (1.33 − 1.23i)14-s + (−0.00183 − 0.00125i)15-s + (−0.314 + 0.394i)16-s + (−1.41 − 0.213i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.169 + 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.169 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.424898 - 0.504471i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.424898 - 0.504471i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-6.54 - 0.341i)T \) |
| good | 2 | \( 1 + (0.483 + 2.11i)T + (-1.80 + 0.867i)T^{2} \) |
| 3 | \( 1 + (0.240 + 0.222i)T + (0.224 + 2.99i)T^{2} \) |
| 5 | \( 1 + (-0.0260 + 0.00392i)T + (4.77 - 1.47i)T^{2} \) |
| 7 | \( 1 + (-1.56 - 2.71i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-3.36 - 1.62i)T + (6.85 + 8.60i)T^{2} \) |
| 13 | \( 1 + (1.78 - 4.55i)T + (-9.52 - 8.84i)T^{2} \) |
| 17 | \( 1 + (5.84 + 0.881i)T + (16.2 + 5.01i)T^{2} \) |
| 19 | \( 1 + (-0.296 + 3.95i)T + (-18.7 - 2.83i)T^{2} \) |
| 23 | \( 1 + (-0.284 + 0.193i)T + (8.40 - 21.4i)T^{2} \) |
| 29 | \( 1 + (-0.971 + 0.901i)T + (2.16 - 28.9i)T^{2} \) |
| 31 | \( 1 + (-2.06 - 0.638i)T + (25.6 + 17.4i)T^{2} \) |
| 37 | \( 1 + (-5.01 + 8.69i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (0.431 + 1.89i)T + (-36.9 + 17.7i)T^{2} \) |
| 47 | \( 1 + (10.0 - 4.82i)T + (29.3 - 36.7i)T^{2} \) |
| 53 | \( 1 + (2.12 + 5.41i)T + (-38.8 + 36.0i)T^{2} \) |
| 59 | \( 1 + (-0.107 + 0.134i)T + (-13.1 - 57.5i)T^{2} \) |
| 61 | \( 1 + (-5.31 + 1.63i)T + (50.4 - 34.3i)T^{2} \) |
| 67 | \( 1 + (0.0822 - 1.09i)T + (-66.2 - 9.98i)T^{2} \) |
| 71 | \( 1 + (4.37 + 2.98i)T + (25.9 + 66.0i)T^{2} \) |
| 73 | \( 1 + (2.70 - 6.88i)T + (-53.5 - 49.6i)T^{2} \) |
| 79 | \( 1 + (-2.70 - 4.68i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (1.49 + 1.38i)T + (6.20 + 82.7i)T^{2} \) |
| 89 | \( 1 + (-2.93 - 2.72i)T + (6.65 + 88.7i)T^{2} \) |
| 97 | \( 1 + (8.11 + 3.90i)T + (60.4 + 75.8i)T^{2} \) |
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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
−15.56566394026028095318945337804, −14.40634159276742343817091985371, −12.82747367250164263272577826983, −11.69777779256258156414088205696, −11.43552282252419398297077696695, −9.425772754906663838916920454948, −9.009732890816113394393154452052, −6.61441496055964137251381488756, −4.29120729359600041872843454959, −2.14658750650857661649073908884,
4.61792356286841438820273108404, 6.13201291478136233402654405737, 7.56331372349205409795934001001, 8.396734749976490527397574152036, 10.09642815100240970325828759190, 11.38598978150519041666466092450, 13.42290768950974482724241736018, 14.28533390759817113197388777809, 15.33408216153283057883264273465, 16.47971891751574482271534630441
