| L(s) = 1 | + (−1.32 + 0.488i)2-s + (0.526 − 1.65i)3-s + (1.52 − 1.29i)4-s + (−0.114 − 0.136i)5-s + (0.108 + 2.44i)6-s + (2.42 − 0.881i)7-s + (−1.38 + 2.46i)8-s + (−2.44 − 1.73i)9-s + (0.219 + 0.125i)10-s + (0.599 − 0.714i)11-s + (−1.33 − 3.19i)12-s + (0.368 − 0.0649i)13-s + (−2.78 + 2.35i)14-s + (−0.285 + 0.117i)15-s + (0.637 − 3.94i)16-s + (−2.26 − 3.91i)17-s + ⋯ |
| L(s) = 1 | + (−0.938 + 0.345i)2-s + (0.303 − 0.952i)3-s + (0.761 − 0.648i)4-s + (−0.0513 − 0.0611i)5-s + (0.0441 + 0.999i)6-s + (0.914 − 0.333i)7-s + (−0.490 + 0.871i)8-s + (−0.815 − 0.578i)9-s + (0.0692 + 0.0396i)10-s + (0.180 − 0.215i)11-s + (−0.386 − 0.922i)12-s + (0.102 − 0.0180i)13-s + (−0.743 + 0.628i)14-s + (−0.0738 + 0.0303i)15-s + (0.159 − 0.987i)16-s + (−0.548 − 0.950i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.410 + 0.911i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.410 + 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.767991 - 0.496345i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.767991 - 0.496345i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.32 - 0.488i)T \) |
| 3 | \( 1 + (-0.526 + 1.65i)T \) |
| good | 5 | \( 1 + (0.114 + 0.136i)T + (-0.868 + 4.92i)T^{2} \) |
| 7 | \( 1 + (-2.42 + 0.881i)T + (5.36 - 4.49i)T^{2} \) |
| 11 | \( 1 + (-0.599 + 0.714i)T + (-1.91 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.368 + 0.0649i)T + (12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (2.26 + 3.91i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.97 + 1.14i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.40 - 1.60i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-1.58 - 0.279i)T + (27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (-8.17 - 2.97i)T + (23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + (3.82 - 2.20i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.238 - 1.35i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (3.28 - 3.91i)T + (-7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (7.70 - 2.80i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + 9.07iT - 53T^{2} \) |
| 59 | \( 1 + (-9.44 - 11.2i)T + (-10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-5.07 - 13.9i)T + (-46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-1.72 + 0.303i)T + (62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-2.77 - 4.80i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (4.77 - 8.26i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.704 - 3.99i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-15.0 - 2.65i)T + (77.9 + 28.3i)T^{2} \) |
| 89 | \( 1 + (-8.16 + 14.1i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (8.94 + 7.50i)T + (16.8 + 95.5i)T^{2} \) |
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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
−11.79023769019943175237995411198, −11.31234501420076635339684455630, −10.09306721727589047056481071422, −8.768822496281462135921062662594, −8.269153512231680425076288296684, −7.17777319732848024859677226523, −6.44155132111951514634387457444, −4.94584102818651580255038525243, −2.65648207418455771518269322994, −1.11273308996361471187842821506,
2.07576432251330197439594825073, 3.60119482866874564368088363549, 4.92789240948492298061690356597, 6.51551767362773042062293285218, 8.019470716144030678382780975145, 8.613703038891498347978559839853, 9.536473572495425699558768224481, 10.59403933609996644042657540206, 11.16044224670480796369297303026, 12.11692271795089021970942146042
