L(s) = 1 | + (−0.959 − 0.281i)2-s + (0.755 + 0.654i)3-s + (0.841 + 0.540i)4-s + (−0.0416 + 0.289i)5-s + (−0.540 − 0.841i)6-s + (−2.26 − 1.37i)7-s + (−0.654 − 0.755i)8-s + (0.142 + 0.989i)9-s + (0.121 − 0.266i)10-s + (−0.513 − 1.74i)11-s + (0.281 + 0.959i)12-s + (2.34 + 1.07i)13-s + (1.78 + 1.95i)14-s + (−0.221 + 0.191i)15-s + (0.415 + 0.909i)16-s + (−0.948 + 0.609i)17-s + ⋯ |
L(s) = 1 | + (−0.678 − 0.199i)2-s + (0.436 + 0.378i)3-s + (0.420 + 0.270i)4-s + (−0.0186 + 0.129i)5-s + (−0.220 − 0.343i)6-s + (−0.855 − 0.518i)7-s + (−0.231 − 0.267i)8-s + (0.0474 + 0.329i)9-s + (0.0384 − 0.0841i)10-s + (−0.154 − 0.527i)11-s + (0.0813 + 0.276i)12-s + (0.651 + 0.297i)13-s + (0.477 + 0.521i)14-s + (−0.0570 + 0.0494i)15-s + (0.103 + 0.227i)16-s + (−0.230 + 0.147i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.25783 + 0.0514257i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.25783 + 0.0514257i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.959 + 0.281i)T \) |
| 3 | \( 1 + (-0.755 - 0.654i)T \) |
| 7 | \( 1 + (2.26 + 1.37i)T \) |
| 23 | \( 1 + (-4.54 + 1.52i)T \) |
good | 5 | \( 1 + (0.0416 - 0.289i)T + (-4.79 - 1.40i)T^{2} \) |
| 11 | \( 1 + (0.513 + 1.74i)T + (-9.25 + 5.94i)T^{2} \) |
| 13 | \( 1 + (-2.34 - 1.07i)T + (8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (0.948 - 0.609i)T + (7.06 - 15.4i)T^{2} \) |
| 19 | \( 1 + (-2.71 - 1.74i)T + (7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (-3.78 + 2.43i)T + (12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (-2.26 + 1.96i)T + (4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (-4.55 + 0.654i)T + (35.5 - 10.4i)T^{2} \) |
| 41 | \( 1 + (2.18 + 0.314i)T + (39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (-3.03 - 2.62i)T + (6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 - 9.34iT - 47T^{2} \) |
| 53 | \( 1 + (-10.4 + 4.75i)T + (34.7 - 40.0i)T^{2} \) |
| 59 | \( 1 + (11.1 + 5.10i)T + (38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (-6.30 - 7.27i)T + (-8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (-2.51 + 8.55i)T + (-56.3 - 36.2i)T^{2} \) |
| 71 | \( 1 + (12.9 + 3.80i)T + (59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (3.01 - 4.68i)T + (-30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (-6.36 - 2.90i)T + (51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (-1.09 - 7.60i)T + (-79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (-6.28 + 7.25i)T + (-12.6 - 88.0i)T^{2} \) |
| 97 | \( 1 + (1.34 - 9.33i)T + (-93.0 - 27.3i)T^{2} \) |
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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
−9.954682592240068405293467679123, −9.237906683941354568363583296468, −8.545344774610610166631635683522, −7.65852932528893668076369354369, −6.75789580616791118940098630332, −5.94904854048244357962756605628, −4.50733138481857632578187806337, −3.44975421701134755789714954246, −2.71623468766907963787853694995, −0.991316620918255724504426386225,
0.974754203990659380648814446039, 2.49044955623177807976368206544, 3.32614408382733317737370182924, 4.87357958583827758589610161328, 5.94447495710687816495957149602, 6.84259877447552448970393051114, 7.42067350666569719903434106551, 8.656644054387180951096177082282, 8.903060654377729272160065919813, 9.873247964366957101944161733127