| L(s) = 1 | + (−4 − 6.92i)2-s + (26.8 + 46.5i)3-s + (−31.9 + 55.4i)4-s + (253. + 439. i)5-s + (214. − 372. i)6-s − 1.03e3·7-s + 511.·8-s + (−348. + 604. i)9-s + (2.02e3 − 3.51e3i)10-s − 7.90e3·11-s − 3.43e3·12-s + (3.11e3 − 5.40e3i)13-s + (4.13e3 + 7.15e3i)14-s + (−1.36e4 + 2.35e4i)15-s + (−2.04e3 − 3.54e3i)16-s + (−2.60e3 − 4.50e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.353 − 0.612i)2-s + (0.574 + 0.994i)3-s + (−0.249 + 0.433i)4-s + (0.907 + 1.57i)5-s + (0.406 − 0.703i)6-s − 1.13·7-s + 0.353·8-s + (−0.159 + 0.276i)9-s + (0.641 − 1.11i)10-s − 1.79·11-s − 0.574·12-s + (0.393 − 0.681i)13-s + (0.402 + 0.697i)14-s + (−1.04 + 1.80i)15-s + (−0.125 − 0.216i)16-s + (−0.128 − 0.222i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.656 - 0.754i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.656 - 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.518007 + 1.13782i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.518007 + 1.13782i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (4 + 6.92i)T \) |
| 19 | \( 1 + (-2.39e4 - 1.78e4i)T \) |
| good | 3 | \( 1 + (-26.8 - 46.5i)T + (-1.09e3 + 1.89e3i)T^{2} \) |
| 5 | \( 1 + (-253. - 439. i)T + (-3.90e4 + 6.76e4i)T^{2} \) |
| 7 | \( 1 + 1.03e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 7.90e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + (-3.11e3 + 5.40e3i)T + (-3.13e7 - 5.43e7i)T^{2} \) |
| 17 | \( 1 + (2.60e3 + 4.50e3i)T + (-2.05e8 + 3.55e8i)T^{2} \) |
| 23 | \( 1 + (4.84e4 - 8.39e4i)T + (-1.70e9 - 2.94e9i)T^{2} \) |
| 29 | \( 1 + (-5.21e4 + 9.03e4i)T + (-8.62e9 - 1.49e10i)T^{2} \) |
| 31 | \( 1 + 1.08e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 3.24e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (-1.03e5 - 1.79e5i)T + (-9.73e10 + 1.68e11i)T^{2} \) |
| 43 | \( 1 + (-6.72e4 - 1.16e5i)T + (-1.35e11 + 2.35e11i)T^{2} \) |
| 47 | \( 1 + (4.01e5 - 6.95e5i)T + (-2.53e11 - 4.38e11i)T^{2} \) |
| 53 | \( 1 + (3.51e5 - 6.09e5i)T + (-5.87e11 - 1.01e12i)T^{2} \) |
| 59 | \( 1 + (-1.10e6 - 1.90e6i)T + (-1.24e12 + 2.15e12i)T^{2} \) |
| 61 | \( 1 + (2.96e5 - 5.13e5i)T + (-1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-1.49e6 + 2.59e6i)T + (-3.03e12 - 5.24e12i)T^{2} \) |
| 71 | \( 1 + (-6.61e5 - 1.14e6i)T + (-4.54e12 + 7.87e12i)T^{2} \) |
| 73 | \( 1 + (-1.22e6 - 2.11e6i)T + (-5.52e12 + 9.56e12i)T^{2} \) |
| 79 | \( 1 + (-5.47e5 - 9.47e5i)T + (-9.60e12 + 1.66e13i)T^{2} \) |
| 83 | \( 1 - 1.38e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + (-9.84e5 + 1.70e6i)T + (-2.21e13 - 3.83e13i)T^{2} \) |
| 97 | \( 1 + (5.43e6 + 9.41e6i)T + (-4.03e13 + 6.99e13i)T^{2} \) |
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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
−15.33121624716622271066217459162, −13.95354577023901206454847516997, −13.03394108690068009367517525073, −10.98501966876970110263638123814, −9.959951266465783140691259805703, −9.732954989295761792680893476034, −7.67076933729077432531569119096, −5.85848351942883704170711289613, −3.38109322399997939734763761078, −2.69219503831087371435626878721,
0.54856515156220229765984334076, 2.18943623851267097219908406280, 5.10157406742397014002505319911, 6.48880838038538185037930730727, 8.013666757822468964307610653254, 8.972569823449737437908435350012, 10.12197736427140581798923883149, 12.70755162848255472049111776547, 13.10487319998610043344154398382, 13.97381922638900622782475907959
