| L(s) = 1 | + (−0.957 + 1.04i)2-s + (0.760 − 1.55i)3-s + (−0.166 − 1.99i)4-s + (−1.70 − 1.44i)5-s + (0.891 + 2.28i)6-s − 1.06·7-s + (2.23 + 1.73i)8-s + (−1.84 − 2.36i)9-s + (3.13 − 0.391i)10-s + (−3.04 + 2.21i)11-s + (−3.22 − 1.25i)12-s + (0.659 − 0.908i)13-s + (1.01 − 1.10i)14-s + (−3.54 + 1.55i)15-s + (−3.94 + 0.663i)16-s + (−0.888 − 2.73i)17-s + ⋯ |
| L(s) = 1 | + (−0.677 + 0.735i)2-s + (0.438 − 0.898i)3-s + (−0.0831 − 0.996i)4-s + (−0.762 − 0.646i)5-s + (0.364 + 0.931i)6-s − 0.402·7-s + (0.789 + 0.613i)8-s + (−0.614 − 0.788i)9-s + (0.992 − 0.123i)10-s + (−0.917 + 0.666i)11-s + (−0.931 − 0.362i)12-s + (0.183 − 0.251i)13-s + (0.272 − 0.296i)14-s + (−0.915 + 0.401i)15-s + (−0.986 + 0.165i)16-s + (−0.215 − 0.663i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.650 + 0.759i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.650 + 0.759i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.204037 - 0.443749i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.204037 - 0.443749i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.957 - 1.04i)T \) |
| 3 | \( 1 + (-0.760 + 1.55i)T \) |
| 5 | \( 1 + (1.70 + 1.44i)T \) |
| good | 7 | \( 1 + 1.06T + 7T^{2} \) |
| 11 | \( 1 + (3.04 - 2.21i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (-0.659 + 0.908i)T + (-4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (0.888 + 2.73i)T + (-13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (4.04 - 1.31i)T + (15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (4.00 + 5.50i)T + (-7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-2.10 - 0.685i)T + (23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-0.361 + 0.117i)T + (25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-6.98 + 9.61i)T + (-11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (-2.81 + 3.87i)T + (-12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 9.24T + 43T^{2} \) |
| 47 | \( 1 + (7.50 + 2.44i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-1.84 + 5.67i)T + (-42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (-5.06 - 3.67i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-2.73 + 1.98i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-2.81 - 8.65i)T + (-54.2 + 39.3i)T^{2} \) |
| 71 | \( 1 + (1.55 - 4.79i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (0.951 + 1.30i)T + (-22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (13.0 + 4.22i)T + (63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-14.7 + 4.80i)T + (67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (-4.85 - 6.68i)T + (-27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (11.0 + 3.58i)T + (78.4 + 57.0i)T^{2} \) |
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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
−11.39857738072722917052884759466, −10.25386842591093254200309444119, −9.162596535270302248813991496585, −8.325890692808917325031159413486, −7.68484882040739486476341014249, −6.80620237771317319378145050370, −5.68001713128991394571888351189, −4.31165901112148091503964488825, −2.32437306159683067931528360156, −0.40456311495382999552221235655,
2.58268749308742203853703563608, 3.51001864930537156659543534856, 4.47328195090500913979451773384, 6.30380111982899978403854903592, 7.84117914774357458029426330881, 8.289467590604080238613797871270, 9.434255523679332448062365684210, 10.30154415519271718382847194184, 10.97989609379235819820788485910, 11.62366057941956114699111497798
