| L(s) = 1 | + (−1.27 − 0.616i)2-s + (−1.71 + 0.267i)3-s + (1.23 + 1.57i)4-s + (1.18 − 3.49i)5-s + (2.34 + 0.714i)6-s + (2.38 − 3.10i)7-s + (−0.607 − 2.76i)8-s + (2.85 − 0.916i)9-s + (−3.66 + 3.71i)10-s + (5.14 + 0.337i)11-s + (−2.54 − 2.35i)12-s + (−0.769 + 0.877i)13-s + (−4.94 + 2.48i)14-s + (−1.09 + 6.30i)15-s + (−0.930 + 3.89i)16-s + (−0.383 − 0.383i)17-s + ⋯ |
| L(s) = 1 | + (−0.899 − 0.436i)2-s + (−0.987 + 0.154i)3-s + (0.619 + 0.785i)4-s + (0.530 − 1.56i)5-s + (0.956 + 0.291i)6-s + (0.900 − 1.17i)7-s + (−0.214 − 0.976i)8-s + (0.952 − 0.305i)9-s + (−1.15 + 1.17i)10-s + (1.55 + 0.101i)11-s + (−0.733 − 0.679i)12-s + (−0.213 + 0.243i)13-s + (−1.32 + 0.663i)14-s + (−0.282 + 1.62i)15-s + (−0.232 + 0.972i)16-s + (−0.0929 − 0.0929i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.225 + 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.225 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.583418 - 0.733543i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.583418 - 0.733543i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.27 + 0.616i)T \) |
| 3 | \( 1 + (1.71 - 0.267i)T \) |
| good | 5 | \( 1 + (-1.18 + 3.49i)T + (-3.96 - 3.04i)T^{2} \) |
| 7 | \( 1 + (-2.38 + 3.10i)T + (-1.81 - 6.76i)T^{2} \) |
| 11 | \( 1 + (-5.14 - 0.337i)T + (10.9 + 1.43i)T^{2} \) |
| 13 | \( 1 + (0.769 - 0.877i)T + (-1.69 - 12.8i)T^{2} \) |
| 17 | \( 1 + (0.383 + 0.383i)T + 17iT^{2} \) |
| 19 | \( 1 + (1.01 - 5.12i)T + (-17.5 - 7.27i)T^{2} \) |
| 23 | \( 1 + (-2.93 + 2.25i)T + (5.95 - 22.2i)T^{2} \) |
| 29 | \( 1 + (-2.16 + 4.39i)T + (-17.6 - 23.0i)T^{2} \) |
| 31 | \( 1 + (-8.18 - 4.72i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.51 + 7.62i)T + (-34.1 + 14.1i)T^{2} \) |
| 41 | \( 1 + (-2.73 + 2.09i)T + (10.6 - 39.6i)T^{2} \) |
| 43 | \( 1 + (0.376 - 5.75i)T + (-42.6 - 5.61i)T^{2} \) |
| 47 | \( 1 + (0.0370 - 0.138i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (4.12 - 6.17i)T + (-20.2 - 48.9i)T^{2} \) |
| 59 | \( 1 + (1.44 - 4.25i)T + (-46.8 - 35.9i)T^{2} \) |
| 61 | \( 1 + (5.25 + 2.59i)T + (37.1 + 48.3i)T^{2} \) |
| 67 | \( 1 + (-0.782 - 11.9i)T + (-66.4 + 8.74i)T^{2} \) |
| 71 | \( 1 + (3.02 + 7.31i)T + (-50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (-2.04 + 4.93i)T + (-51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (4.71 + 1.26i)T + (68.4 + 39.5i)T^{2} \) |
| 83 | \( 1 + (8.36 - 2.83i)T + (65.8 - 50.5i)T^{2} \) |
| 89 | \( 1 + (10.4 - 4.31i)T + (62.9 - 62.9i)T^{2} \) |
| 97 | \( 1 + (3.86 - 2.22i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.41299583457853244759300854247, −9.668566212074011286599899491920, −8.908300833062697774626393076734, −7.990268097110779177747008125939, −6.93872529146967173287168922725, −5.98850650056568329118463113728, −4.54391548065333613181375543562, −4.15498139545688867182811083458, −1.52979619597668935771684891160, −0.980350313034532165479739864045,
1.55065714786702805866825086232, 2.73137268968993520235596620795, 4.84746856990754637474645417921, 5.87560817837802023057163235570, 6.57289228235701439507816069324, 7.08316117321067535298195639227, 8.326763443907950775962372086841, 9.338666281690685643016185181159, 10.09622065044720527282010613894, 11.13200081095339489259631936842
