L(s) = 1 | + (0.796 − 1.53i)3-s + (0.230 + 0.398i)5-s + (−0.0665 − 2.64i)7-s + (−1.73 − 2.45i)9-s + (−1.82 + 3.15i)11-s + (0.730 − 1.26i)13-s + (0.796 − 0.0363i)15-s + (−1.86 − 3.23i)17-s + (2.02 − 3.51i)19-s + (−4.12 − 2.00i)21-s + (0.566 + 0.981i)23-s + (2.39 − 4.14i)25-s + (−5.14 + 0.708i)27-s + (−4.48 − 7.77i)29-s + 0.514·31-s + ⋯ |
L(s) = 1 | + (0.460 − 0.887i)3-s + (0.102 + 0.178i)5-s + (−0.0251 − 0.999i)7-s + (−0.576 − 0.816i)9-s + (−0.549 + 0.952i)11-s + (0.202 − 0.350i)13-s + (0.205 − 0.00938i)15-s + (−0.452 − 0.784i)17-s + (0.465 − 0.805i)19-s + (−0.899 − 0.437i)21-s + (0.118 + 0.204i)23-s + (0.478 − 0.829i)25-s + (−0.990 + 0.136i)27-s + (−0.833 − 1.44i)29-s + 0.0924·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.638 + 0.769i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.638 + 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.485222920\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.485222920\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.796 + 1.53i)T \) |
| 7 | \( 1 + (0.0665 + 2.64i)T \) |
good | 5 | \( 1 + (-0.230 - 0.398i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (1.82 - 3.15i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.730 + 1.26i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (1.86 + 3.23i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.02 + 3.51i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.566 - 0.981i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (4.48 + 7.77i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 0.514T + 31T^{2} \) |
| 37 | \( 1 + (4.55 - 7.88i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (0.472 - 0.819i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (4.66 + 8.07i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 2.32T + 47T^{2} \) |
| 53 | \( 1 + (-6.21 - 10.7i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 - 12.8T + 59T^{2} \) |
| 61 | \( 1 - 12.0T + 61T^{2} \) |
| 67 | \( 1 - 2.32T + 67T^{2} \) |
| 71 | \( 1 + 1.67T + 71T^{2} \) |
| 73 | \( 1 + (6.62 + 11.4i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 - 5.00T + 79T^{2} \) |
| 83 | \( 1 + (3.32 + 5.75i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (1.36 - 2.36i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (5.59 + 9.68i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.731110243448423871396854954894, −8.717574429256265248203126541449, −7.82842583866908195924399655605, −7.13441404492412241055501512863, −6.64517518483221519922721365052, −5.34905806017071339235062978138, −4.28111835877467987750646258256, −3.07909097557052223190310110335, −2.10317197255407508664578916017, −0.62034208852490406629821499416,
1.91767305293625226963001913516, 3.11528963097026558462445110712, 3.88964880745718821979782110897, 5.33002620918096334115765300154, 5.54162164774283750785680254642, 6.87070760134070692327946040449, 8.228728891077519426617115939649, 8.607414066343825346063273821200, 9.329856283384574363270827683250, 10.18723838706135580222466634846