| L(s) = 1 | + (−0.939 + 0.342i)2-s + (−1.04 − 1.38i)3-s + (0.766 − 0.642i)4-s + (0.108 + 0.617i)5-s + (1.45 + 0.940i)6-s + (0.766 + 0.642i)7-s + (−0.500 + 0.866i)8-s + (−0.815 + 2.88i)9-s + (−0.313 − 0.542i)10-s + (−0.0434 + 0.246i)11-s + (−1.68 − 0.386i)12-s + (3.98 + 1.45i)13-s + (−0.939 − 0.342i)14-s + (0.739 − 0.795i)15-s + (0.173 − 0.984i)16-s + (0.0870 + 0.150i)17-s + ⋯ |
| L(s) = 1 | + (−0.664 + 0.241i)2-s + (−0.603 − 0.797i)3-s + (0.383 − 0.321i)4-s + (0.0486 + 0.276i)5-s + (0.593 + 0.383i)6-s + (0.289 + 0.242i)7-s + (−0.176 + 0.306i)8-s + (−0.271 + 0.962i)9-s + (−0.0991 − 0.171i)10-s + (−0.0130 + 0.0742i)11-s + (−0.487 − 0.111i)12-s + (1.10 + 0.402i)13-s + (−0.251 − 0.0914i)14-s + (0.190 − 0.205i)15-s + (0.0434 − 0.246i)16-s + (0.0211 + 0.0365i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0348i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0348i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.880191 - 0.0153432i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.880191 - 0.0153432i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.939 - 0.342i)T \) |
| 3 | \( 1 + (1.04 + 1.38i)T \) |
| 7 | \( 1 + (-0.766 - 0.642i)T \) |
| good | 5 | \( 1 + (-0.108 - 0.617i)T + (-4.69 + 1.71i)T^{2} \) |
| 11 | \( 1 + (0.0434 - 0.246i)T + (-10.3 - 3.76i)T^{2} \) |
| 13 | \( 1 + (-3.98 - 1.45i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-0.0870 - 0.150i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.22 + 2.12i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.0380 + 0.0319i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (-0.635 + 0.231i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (-5.73 + 4.81i)T + (5.38 - 30.5i)T^{2} \) |
| 37 | \( 1 + (-2.59 - 4.50i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-11.3 - 4.14i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-0.237 + 1.34i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-0.339 - 0.285i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + 12.0T + 53T^{2} \) |
| 59 | \( 1 + (-0.847 - 4.80i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (2.13 + 1.79i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (3.13 + 1.14i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (2.14 + 3.71i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (4.93 - 8.54i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.12 + 1.86i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-5.81 + 2.11i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (-3.53 + 6.12i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (2.64 - 15.0i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−11.25178426981754904521664898966, −10.66217597536045543661719579616, −9.440779201808952120950726703022, −8.410413720682314146068328117917, −7.64423717478983407362697057741, −6.56503564335252175970044995582, −5.98365847536767597251956336158, −4.65915801822237368572504789076, −2.64712349851005546416659957992, −1.17125434997677926028711906738,
1.06425874539303324001085827736, 3.18837348496177616236648340716, 4.35331110467483203635562980464, 5.55877607665463289332495634299, 6.53831832277671753809657767269, 7.86625236768918847858880158124, 8.808030031962588574214448910512, 9.577257999303747050888410160961, 10.65652509160657801015357415883, 10.98167375635548391608161451491
