| L(s) = 1 | + (−0.907 + 0.419i)2-s + (0.947 + 0.319i)3-s + (0.647 − 0.762i)4-s + (−1.91 − 1.15i)5-s + (−0.994 + 0.108i)6-s + (0.0631 − 1.16i)7-s + (−0.267 + 0.963i)8-s + (0.796 + 0.605i)9-s + (2.22 + 0.241i)10-s + (0.268 − 1.63i)11-s + (0.856 − 0.515i)12-s + (2.88 − 2.19i)13-s + (0.431 + 1.08i)14-s + (−1.44 − 1.70i)15-s + (−0.161 − 0.986i)16-s + (−0.179 − 3.30i)17-s + ⋯ |
| L(s) = 1 | + (−0.641 + 0.296i)2-s + (0.547 + 0.184i)3-s + (0.323 − 0.381i)4-s + (−0.856 − 0.515i)5-s + (−0.405 + 0.0441i)6-s + (0.0238 − 0.440i)7-s + (−0.0945 + 0.340i)8-s + (0.265 + 0.201i)9-s + (0.702 + 0.0764i)10-s + (0.0808 − 0.493i)11-s + (0.247 − 0.148i)12-s + (0.799 − 0.607i)13-s + (0.115 + 0.289i)14-s + (−0.373 − 0.439i)15-s + (−0.0404 − 0.246i)16-s + (−0.0434 − 0.802i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.710 + 0.704i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.710 + 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.931279 - 0.383390i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.931279 - 0.383390i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.907 - 0.419i)T \) |
| 3 | \( 1 + (-0.947 - 0.319i)T \) |
| 59 | \( 1 + (-7.48 - 1.70i)T \) |
| good | 5 | \( 1 + (1.91 + 1.15i)T + (2.34 + 4.41i)T^{2} \) |
| 7 | \( 1 + (-0.0631 + 1.16i)T + (-6.95 - 0.756i)T^{2} \) |
| 11 | \( 1 + (-0.268 + 1.63i)T + (-10.4 - 3.51i)T^{2} \) |
| 13 | \( 1 + (-2.88 + 2.19i)T + (3.47 - 12.5i)T^{2} \) |
| 17 | \( 1 + (0.179 + 3.30i)T + (-16.9 + 1.83i)T^{2} \) |
| 19 | \( 1 + (-3.45 + 3.26i)T + (1.02 - 18.9i)T^{2} \) |
| 23 | \( 1 + (-0.109 + 0.0241i)T + (20.8 - 9.65i)T^{2} \) |
| 29 | \( 1 + (1.29 + 0.601i)T + (18.7 + 22.1i)T^{2} \) |
| 31 | \( 1 + (-1.66 - 1.57i)T + (1.67 + 30.9i)T^{2} \) |
| 37 | \( 1 + (-0.961 - 3.46i)T + (-31.7 + 19.0i)T^{2} \) |
| 41 | \( 1 + (0.297 + 0.0655i)T + (37.2 + 17.2i)T^{2} \) |
| 43 | \( 1 + (0.582 + 3.55i)T + (-40.7 + 13.7i)T^{2} \) |
| 47 | \( 1 + (-1.19 + 0.720i)T + (22.0 - 41.5i)T^{2} \) |
| 53 | \( 1 + (1.98 - 0.215i)T + (51.7 - 11.3i)T^{2} \) |
| 61 | \( 1 + (9.95 - 4.60i)T + (39.4 - 46.4i)T^{2} \) |
| 67 | \( 1 + (-1.01 + 3.64i)T + (-57.4 - 34.5i)T^{2} \) |
| 71 | \( 1 + (8.76 - 5.27i)T + (33.2 - 62.7i)T^{2} \) |
| 73 | \( 1 + (-2.36 - 5.92i)T + (-52.9 + 50.2i)T^{2} \) |
| 79 | \( 1 + (8.73 - 2.94i)T + (62.8 - 47.8i)T^{2} \) |
| 83 | \( 1 + (1.97 - 2.91i)T + (-30.7 - 77.1i)T^{2} \) |
| 89 | \( 1 + (-8.16 - 3.77i)T + (57.6 + 67.8i)T^{2} \) |
| 97 | \( 1 + (6.01 - 15.0i)T + (-70.4 - 66.7i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.24522748654230677011377120113, −10.34294521683905027766890002246, −9.282177466109428665956758081217, −8.509395375605112200303481874144, −7.79813455342209721829550113495, −6.90905148429750997785116797890, −5.46884710961378235509935193046, −4.22651218426640258379946699064, −3.01103483314680768867317968915, −0.872511738862637223184846913021,
1.74310890947377713261367244323, 3.21488733126713580857880537759, 4.14420102715610384339666944765, 6.02033699495020265813618257277, 7.16949315764714515680546969563, 7.910859123884951632768082167388, 8.765281713359029911139067717338, 9.645659767377085922110661008448, 10.68251066741471680333191475562, 11.55726449490046026978942030869
