| L(s) = 1 | + (7.51 − 2.73i)2-s + (−19.5 − 7.11i)3-s + (49.0 − 41.1i)4-s + (69.5 + 394. i)5-s − 166.·6-s + (−65.8 − 373. i)7-s + (256. − 443. i)8-s + (−1.34e3 − 1.12e3i)9-s + (1.60e3 + 2.77e3i)10-s + (−2.61e3 + 4.52e3i)11-s + (−1.25e3 + 455. i)12-s + (1.58e3 − 1.33e3i)13-s + (−1.51e3 − 2.62e3i)14-s + (1.44e3 − 8.20e3i)15-s + (711. − 4.03e3i)16-s + (−2.42e4 − 2.03e4i)17-s + ⋯ |
| L(s) = 1 | + (0.664 − 0.241i)2-s + (−0.417 − 0.152i)3-s + (0.383 − 0.321i)4-s + (0.248 + 1.41i)5-s − 0.314·6-s + (−0.0725 − 0.411i)7-s + (0.176 − 0.306i)8-s + (−0.614 − 0.515i)9-s + (0.506 + 0.877i)10-s + (−0.591 + 1.02i)11-s + (−0.208 + 0.0760i)12-s + (0.200 − 0.167i)13-s + (−0.147 − 0.255i)14-s + (0.110 − 0.627i)15-s + (0.0434 − 0.246i)16-s + (−1.19 − 1.00i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.926 - 0.377i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.926 - 0.377i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.0748705 + 0.382387i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0748705 + 0.382387i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-7.51 + 2.73i)T \) |
| 37 | \( 1 + (-2.87e5 + 1.11e5i)T \) |
| good | 3 | \( 1 + (19.5 + 7.11i)T + (1.67e3 + 1.40e3i)T^{2} \) |
| 5 | \( 1 + (-69.5 - 394. i)T + (-7.34e4 + 2.67e4i)T^{2} \) |
| 7 | \( 1 + (65.8 + 373. i)T + (-7.73e5 + 2.81e5i)T^{2} \) |
| 11 | \( 1 + (2.61e3 - 4.52e3i)T + (-9.74e6 - 1.68e7i)T^{2} \) |
| 13 | \( 1 + (-1.58e3 + 1.33e3i)T + (1.08e7 - 6.17e7i)T^{2} \) |
| 17 | \( 1 + (2.42e4 + 2.03e4i)T + (7.12e7 + 4.04e8i)T^{2} \) |
| 19 | \( 1 + (2.60e4 + 9.46e3i)T + (6.84e8 + 5.74e8i)T^{2} \) |
| 23 | \( 1 + (-1.55e4 - 2.68e4i)T + (-1.70e9 + 2.94e9i)T^{2} \) |
| 29 | \( 1 + (1.27e5 - 2.21e5i)T + (-8.62e9 - 1.49e10i)T^{2} \) |
| 31 | \( 1 + 2.90e5T + 2.75e10T^{2} \) |
| 41 | \( 1 + (3.76e5 - 3.15e5i)T + (3.38e10 - 1.91e11i)T^{2} \) |
| 43 | \( 1 + 2.40e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + (1.96e5 + 3.40e5i)T + (-2.53e11 + 4.38e11i)T^{2} \) |
| 53 | \( 1 + (1.48e5 - 8.42e5i)T + (-1.10e12 - 4.01e11i)T^{2} \) |
| 59 | \( 1 + (-1.63e5 + 9.27e5i)T + (-2.33e12 - 8.51e11i)T^{2} \) |
| 61 | \( 1 + (-1.02e6 + 8.59e5i)T + (5.45e11 - 3.09e12i)T^{2} \) |
| 67 | \( 1 + (6.68e5 + 3.79e6i)T + (-5.69e12 + 2.07e12i)T^{2} \) |
| 71 | \( 1 + (2.35e6 + 8.55e5i)T + (6.96e12 + 5.84e12i)T^{2} \) |
| 73 | \( 1 - 5.32e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + (1.92e5 + 1.09e6i)T + (-1.80e13 + 6.56e12i)T^{2} \) |
| 83 | \( 1 + (-7.07e6 - 5.93e6i)T + (4.71e12 + 2.67e13i)T^{2} \) |
| 89 | \( 1 + (2.16e6 - 1.22e7i)T + (-4.15e13 - 1.51e13i)T^{2} \) |
| 97 | \( 1 + (7.49e5 + 1.29e6i)T + (-4.03e13 + 6.99e13i)T^{2} \) |
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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
−13.56404872717863128056281696007, −12.59439585237252791583954770339, −11.15568342412260940664651502224, −10.79590861746968249347117923903, −9.373569653494822376855036557367, −7.21280759177989262647373059046, −6.57415682665465319637395404124, −5.14977954370635497917892975723, −3.44227940775691191010931608812, −2.20057912499640660002424996681,
0.10080551964344067204724181447, 2.14296355369613909411422223393, 4.16891873772899023385634132624, 5.39460229673568699128687260589, 6.09074378641107511274378034961, 8.200429153763231642361137243270, 8.883796006681935556059466862694, 10.71564429617088268867872814741, 11.66282989445871133556455387790, 12.99882588239031588581662061439
