| L(s) = 1 | + (0.907 + 0.419i)2-s + (0.947 − 0.319i)3-s + (0.647 + 0.762i)4-s + (1.95 − 1.17i)5-s + (0.994 + 0.108i)6-s + (0.00134 + 0.0247i)7-s + (0.267 + 0.963i)8-s + (0.796 − 0.605i)9-s + (2.26 − 0.246i)10-s + (−0.227 − 1.38i)11-s + (0.856 + 0.515i)12-s + (−4.57 − 3.47i)13-s + (−0.00916 + 0.0230i)14-s + (1.47 − 1.73i)15-s + (−0.161 + 0.986i)16-s + (−0.409 + 7.54i)17-s + ⋯ |
| L(s) = 1 | + (0.641 + 0.296i)2-s + (0.547 − 0.184i)3-s + (0.323 + 0.381i)4-s + (0.874 − 0.525i)5-s + (0.405 + 0.0441i)6-s + (0.000506 + 0.00934i)7-s + (0.0945 + 0.340i)8-s + (0.265 − 0.201i)9-s + (0.717 − 0.0780i)10-s + (−0.0686 − 0.418i)11-s + (0.247 + 0.148i)12-s + (−1.26 − 0.964i)13-s + (−0.00244 + 0.00614i)14-s + (0.381 − 0.448i)15-s + (−0.0404 + 0.246i)16-s + (−0.0992 + 1.83i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0633i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0633i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.43281 + 0.0770984i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.43281 + 0.0770984i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (6.76 + 3.64i)T \) |
| good | 5 | \( 1 + (-1.95 + 1.17i)T + (2.34 - 4.41i)T^{2} \) |
| 7 | \( 1 + (-0.00134 - 0.0247i)T + (-6.95 + 0.756i)T^{2} \) |
| 11 | \( 1 + (0.227 + 1.38i)T + (-10.4 + 3.51i)T^{2} \) |
| 13 | \( 1 + (4.57 + 3.47i)T + (3.47 + 12.5i)T^{2} \) |
| 17 | \( 1 + (0.409 - 7.54i)T + (-16.9 - 1.83i)T^{2} \) |
| 19 | \( 1 + (-3.23 - 3.06i)T + (1.02 + 18.9i)T^{2} \) |
| 23 | \( 1 + (7.40 + 1.62i)T + (20.8 + 9.65i)T^{2} \) |
| 29 | \( 1 + (-4.18 + 1.93i)T + (18.7 - 22.1i)T^{2} \) |
| 31 | \( 1 + (-7.48 + 7.08i)T + (1.67 - 30.9i)T^{2} \) |
| 37 | \( 1 + (2.90 - 10.4i)T + (-31.7 - 19.0i)T^{2} \) |
| 41 | \( 1 + (9.28 - 2.04i)T + (37.2 - 17.2i)T^{2} \) |
| 43 | \( 1 + (0.628 - 3.83i)T + (-40.7 - 13.7i)T^{2} \) |
| 47 | \( 1 + (4.50 + 2.71i)T + (22.0 + 41.5i)T^{2} \) |
| 53 | \( 1 + (-0.345 - 0.0375i)T + (51.7 + 11.3i)T^{2} \) |
| 61 | \( 1 + (-4.08 - 1.89i)T + (39.4 + 46.4i)T^{2} \) |
| 67 | \( 1 + (2.98 + 10.7i)T + (-57.4 + 34.5i)T^{2} \) |
| 71 | \( 1 + (7.98 + 4.80i)T + (33.2 + 62.7i)T^{2} \) |
| 73 | \( 1 + (-1.26 + 3.16i)T + (-52.9 - 50.2i)T^{2} \) |
| 79 | \( 1 + (1.81 + 0.610i)T + (62.8 + 47.8i)T^{2} \) |
| 83 | \( 1 + (-5.95 - 8.77i)T + (-30.7 + 77.1i)T^{2} \) |
| 89 | \( 1 + (2.80 - 1.29i)T + (57.6 - 67.8i)T^{2} \) |
| 97 | \( 1 + (1.58 + 3.98i)T + (-70.4 + 66.7i)T^{2} \) |
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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
−11.91527024319216392778254562921, −10.21979796540278034966952092263, −9.854932037334785869371789875387, −8.288003347768625216443569416931, −7.969126104709342459649158500697, −6.40749938535361075308882078759, −5.69326388587964869345952508494, −4.54693816511274441617636719190, −3.18621558754751636669868923731, −1.85942872583146234204250697272,
2.12038235271251140653976308188, 2.91914621368469529691333015334, 4.48209172191757673107783291891, 5.32817029989007626008389496226, 6.76838093705790199953937747311, 7.34043543080855355529912392231, 8.989749350221856630427695445583, 9.832813225309469802153073833943, 10.29974478155231703079013021058, 11.73773577988150944863715021068
