| L(s) = 1 | + (0.347 + 1.96i)2-s + (−1.04 + 5.93i)3-s + (−3.75 + 1.36i)4-s + (13.7 + 11.5i)5-s − 12.0·6-s + (−10.6 − 8.94i)7-s + (−4 − 6.92i)8-s + (−8.80 − 3.20i)9-s + (−17.9 + 31.0i)10-s + (−7.16 − 12.4i)11-s + (−4.18 − 23.7i)12-s + (−44.1 + 16.0i)13-s + (13.9 − 24.1i)14-s + (−82.7 + 69.4i)15-s + (12.2 − 10.2i)16-s + (75.5 + 27.5i)17-s + ⋯ |
| L(s) = 1 | + (0.122 + 0.696i)2-s + (−0.201 + 1.14i)3-s + (−0.469 + 0.171i)4-s + (1.22 + 1.02i)5-s − 0.820·6-s + (−0.575 − 0.483i)7-s + (−0.176 − 0.306i)8-s + (−0.326 − 0.118i)9-s + (−0.566 + 0.981i)10-s + (−0.196 − 0.340i)11-s + (−0.100 − 0.571i)12-s + (−0.941 + 0.342i)13-s + (0.265 − 0.460i)14-s + (−1.42 + 1.19i)15-s + (0.191 − 0.160i)16-s + (1.07 + 0.392i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.892 - 0.451i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.892 - 0.451i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.351643 + 1.47364i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.351643 + 1.47364i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.347 - 1.96i)T \) |
| 37 | \( 1 + (225. - 4.36i)T \) |
| good | 3 | \( 1 + (1.04 - 5.93i)T + (-25.3 - 9.23i)T^{2} \) |
| 5 | \( 1 + (-13.7 - 11.5i)T + (21.7 + 123. i)T^{2} \) |
| 7 | \( 1 + (10.6 + 8.94i)T + (59.5 + 337. i)T^{2} \) |
| 11 | \( 1 + (7.16 + 12.4i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (44.1 - 16.0i)T + (1.68e3 - 1.41e3i)T^{2} \) |
| 17 | \( 1 + (-75.5 - 27.5i)T + (3.76e3 + 3.15e3i)T^{2} \) |
| 19 | \( 1 + (-1.81 + 10.3i)T + (-6.44e3 - 2.34e3i)T^{2} \) |
| 23 | \( 1 + (-68.6 + 118. i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-117. - 202. i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 - 182.T + 2.97e4T^{2} \) |
| 41 | \( 1 + (-165. + 60.2i)T + (5.27e4 - 4.43e4i)T^{2} \) |
| 43 | \( 1 - 326.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-88.0 + 152. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (445. - 373. i)T + (2.58e4 - 1.46e5i)T^{2} \) |
| 59 | \( 1 + (400. - 336. i)T + (3.56e4 - 2.02e5i)T^{2} \) |
| 61 | \( 1 + (-248. + 90.5i)T + (1.73e5 - 1.45e5i)T^{2} \) |
| 67 | \( 1 + (595. + 499. i)T + (5.22e4 + 2.96e5i)T^{2} \) |
| 71 | \( 1 + (-95.2 + 540. i)T + (-3.36e5 - 1.22e5i)T^{2} \) |
| 73 | \( 1 - 540.T + 3.89e5T^{2} \) |
| 79 | \( 1 + (-495. - 415. i)T + (8.56e4 + 4.85e5i)T^{2} \) |
| 83 | \( 1 + (937. + 341. i)T + (4.38e5 + 3.67e5i)T^{2} \) |
| 89 | \( 1 + (-1.06e3 + 895. i)T + (1.22e5 - 6.94e5i)T^{2} \) |
| 97 | \( 1 + (-758. + 1.31e3i)T + (-4.56e5 - 7.90e5i)T^{2} \) |
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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
−14.49859500543241395022209719953, −13.94710726365145716443042504611, −12.58066133484519644974415059890, −10.58645652137011844849022821520, −10.17876629826863478242514885568, −9.157611654026230898932442031698, −7.20086650328868259882258151927, −6.11873917520025014201391495663, −4.84106890405912704629939408135, −3.13615687596985777027308506239,
1.05542002266128289655123218407, 2.47898236718575946944707107896, 5.08525880918654524094283198683, 6.14118993133746392144158478134, 7.77934241751274306262579846593, 9.376277793431068233822750995791, 9.987800173400935684317162625743, 11.96775092376520111910985215271, 12.55186042437050814947209787356, 13.23980748912976521663915932233
