| L(s) = 1 | + (2.73 − 0.732i)2-s + (14.6 − 8.48i)3-s + (6.92 − 4i)4-s + (−30.1 − 8.07i)5-s + (33.9 − 33.9i)6-s + (−14.8 − 25.8i)7-s + (15.9 − 16i)8-s + (103. − 179. i)9-s − 88.2·10-s + 172. i·11-s + (67.8 − 117. i)12-s + (73.8 + 19.7i)13-s + (−59.5 − 59.5i)14-s + (−511. + 137. i)15-s + (31.9 − 55.4i)16-s + (100. + 376. i)17-s + ⋯ |
| L(s) = 1 | + (0.683 − 0.183i)2-s + (1.63 − 0.942i)3-s + (0.433 − 0.250i)4-s + (−1.20 − 0.323i)5-s + (0.942 − 0.942i)6-s + (−0.304 − 0.526i)7-s + (0.249 − 0.250i)8-s + (1.27 − 2.21i)9-s − 0.882·10-s + 1.42i·11-s + (0.471 − 0.816i)12-s + (0.436 + 0.117i)13-s + (−0.304 − 0.304i)14-s + (−2.27 + 0.609i)15-s + (0.124 − 0.216i)16-s + (0.349 + 1.30i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.168 + 0.985i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.168 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.53259 - 2.13629i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.53259 - 2.13629i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-2.73 + 0.732i)T \) |
| 37 | \( 1 + (1.36e3 - 143. i)T \) |
| good | 3 | \( 1 + (-14.6 + 8.48i)T + (40.5 - 70.1i)T^{2} \) |
| 5 | \( 1 + (30.1 + 8.07i)T + (541. + 312.5i)T^{2} \) |
| 7 | \( 1 + (14.8 + 25.8i)T + (-1.20e3 + 2.07e3i)T^{2} \) |
| 11 | \( 1 - 172. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + (-73.8 - 19.7i)T + (2.47e4 + 1.42e4i)T^{2} \) |
| 17 | \( 1 + (-100. - 376. i)T + (-7.23e4 + 4.17e4i)T^{2} \) |
| 19 | \( 1 + (-335. - 89.9i)T + (1.12e5 + 6.51e4i)T^{2} \) |
| 23 | \( 1 + (-416. + 416. i)T - 2.79e5iT^{2} \) |
| 29 | \( 1 + (-749. - 749. i)T + 7.07e5iT^{2} \) |
| 31 | \( 1 + (887. + 887. i)T + 9.23e5iT^{2} \) |
| 41 | \( 1 + (680. - 392. i)T + (1.41e6 - 2.44e6i)T^{2} \) |
| 43 | \( 1 + (1.64e3 - 1.64e3i)T - 3.41e6iT^{2} \) |
| 47 | \( 1 + 958.T + 4.87e6T^{2} \) |
| 53 | \( 1 + (-563. + 976. i)T + (-3.94e6 - 6.83e6i)T^{2} \) |
| 59 | \( 1 + (-455. - 1.69e3i)T + (-1.04e7 + 6.05e6i)T^{2} \) |
| 61 | \( 1 + (1.32e3 - 4.96e3i)T + (-1.19e7 - 6.92e6i)T^{2} \) |
| 67 | \( 1 + (6.70e3 - 3.87e3i)T + (1.00e7 - 1.74e7i)T^{2} \) |
| 71 | \( 1 + (2.12e3 + 3.68e3i)T + (-1.27e7 + 2.20e7i)T^{2} \) |
| 73 | \( 1 - 3.28e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + (-3.49e3 - 937. i)T + (3.37e7 + 1.94e7i)T^{2} \) |
| 83 | \( 1 + (-3.90e3 + 6.76e3i)T + (-2.37e7 - 4.11e7i)T^{2} \) |
| 89 | \( 1 + (5.46e3 - 1.46e3i)T + (5.43e7 - 3.13e7i)T^{2} \) |
| 97 | \( 1 + (-4.32e3 + 4.32e3i)T - 8.85e7iT^{2} \) |
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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
−13.44915885968238593744835029742, −12.68644779472225413564526621758, −11.98603534708973790960494724136, −10.13618023948166092346635038153, −8.678439122997966721827467928470, −7.63110411519172091226702425730, −6.85485996948071208664345069818, −4.23948497692583049636261360164, −3.27166169507596165628009703499, −1.48358447193513000137847849953,
3.10890259075671761294720495185, 3.47961024550913755229704538528, 5.11922397445600084286392227662, 7.30882523013976632971155171576, 8.324448546542919709866304017355, 9.272116334143566967543173204345, 10.81572683161071345750306174943, 11.87049581662658570611123859627, 13.56239660143922144116829866380, 14.04612206356124319033970635013
