L(s) = 1 | + (−0.113 − 0.0656i)2-s + (0.332 − 0.0890i)3-s + (−0.991 − 1.71i)4-s + (2.08 − 0.813i)5-s + (−0.0436 − 0.0116i)6-s + (1.39 + 2.40i)7-s + 0.522i·8-s + (−2.49 + 1.44i)9-s + (−0.290 − 0.0442i)10-s + (−3.91 + 1.04i)11-s + (−0.482 − 0.482i)12-s + (0.756 − 3.52i)13-s − 0.365i·14-s + (0.619 − 0.455i)15-s + (−1.94 + 3.37i)16-s + (−0.627 + 2.34i)17-s + ⋯ |
L(s) = 1 | + (−0.0804 − 0.0464i)2-s + (0.191 − 0.0513i)3-s + (−0.495 − 0.858i)4-s + (0.931 − 0.363i)5-s + (−0.0178 − 0.00477i)6-s + (0.525 + 0.910i)7-s + 0.184i·8-s + (−0.831 + 0.480i)9-s + (−0.0917 − 0.0140i)10-s + (−1.18 + 0.316i)11-s + (−0.139 − 0.139i)12-s + (0.209 − 0.977i)13-s − 0.0976i·14-s + (0.159 − 0.117i)15-s + (−0.487 + 0.843i)16-s + (−0.152 + 0.567i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.908 + 0.418i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.908 + 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.881743 - 0.193131i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.881743 - 0.193131i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-2.08 + 0.813i)T \) |
| 13 | \( 1 + (-0.756 + 3.52i)T \) |
good | 2 | \( 1 + (0.113 + 0.0656i)T + (1 + 1.73i)T^{2} \) |
| 3 | \( 1 + (-0.332 + 0.0890i)T + (2.59 - 1.5i)T^{2} \) |
| 7 | \( 1 + (-1.39 - 2.40i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (3.91 - 1.04i)T + (9.52 - 5.5i)T^{2} \) |
| 17 | \( 1 + (0.627 - 2.34i)T + (-14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (-0.491 + 1.83i)T + (-16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (-2.06 - 7.70i)T + (-19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (3.96 + 2.28i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.87 + 3.87i)T - 31iT^{2} \) |
| 37 | \( 1 + (-3.50 + 6.07i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (1.66 + 6.20i)T + (-35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (6.24 + 1.67i)T + (37.2 + 21.5i)T^{2} \) |
| 47 | \( 1 + 0.512T + 47T^{2} \) |
| 53 | \( 1 + (1.32 + 1.32i)T + 53iT^{2} \) |
| 59 | \( 1 + (-2.53 - 0.679i)T + (51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + (-0.641 - 1.11i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.13 - 1.80i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (6.20 + 1.66i)T + (61.4 + 35.5i)T^{2} \) |
| 73 | \( 1 - 9.93iT - 73T^{2} \) |
| 79 | \( 1 - 8.37iT - 79T^{2} \) |
| 83 | \( 1 - 3.17T + 83T^{2} \) |
| 89 | \( 1 + (-1.61 - 6.01i)T + (-77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (-10.1 + 5.88i)T + (48.5 - 84.0i)T^{2} \) |
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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
−14.86158302219870219303896941247, −13.62088875811255312394458120220, −13.01286097862465430391884531570, −11.26296217414437199741886705295, −10.18056436398001231297115388852, −9.069785455010748420992493004729, −8.076563087686087806140083031999, −5.64325359802100540988970506774, −5.29435403307727805906620731664, −2.24766597668038127849594574510,
2.98693751433588238010709913889, 4.79611633433935499902853179618, 6.62466649187397203231335808465, 8.035109401672659491852207279265, 9.093410446033728386887299520975, 10.39700896440475709779908903900, 11.58468567024372803098444052686, 13.08890944855225626763749194075, 13.85561733100869223042742278080, 14.60910332682224290837071347407