L(s) = 1 | + (5.74 − 9.94i)5-s + (17.2 − 6.73i)7-s + (49.6 − 28.6i)11-s − 9.20i·13-s + (14.5 + 25.1i)17-s + (−32.6 − 18.8i)19-s + (−7.27 − 4.19i)23-s + (−3.41 − 5.91i)25-s + 62.3i·29-s + (−48.8 + 28.2i)31-s + (32.1 − 210. i)35-s + (146. − 252. i)37-s + 54.2·41-s − 438.·43-s + (−128. + 222. i)47-s + ⋯ |
L(s) = 1 | + (0.513 − 0.889i)5-s + (0.931 − 0.363i)7-s + (1.36 − 0.785i)11-s − 0.196i·13-s + (0.207 + 0.359i)17-s + (−0.393 − 0.227i)19-s + (−0.0659 − 0.0380i)23-s + (−0.0273 − 0.0473i)25-s + 0.399i·29-s + (−0.283 + 0.163i)31-s + (0.155 − 1.01i)35-s + (0.649 − 1.12i)37-s + 0.206·41-s − 1.55·43-s + (−0.398 + 0.690i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.330 + 0.943i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.330 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.559184586\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.559184586\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-17.2 + 6.73i)T \) |
good | 5 | \( 1 + (-5.74 + 9.94i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (-49.6 + 28.6i)T + (665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + 9.20iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (-14.5 - 25.1i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (32.6 + 18.8i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (7.27 + 4.19i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 - 62.3iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (48.8 - 28.2i)T + (1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-146. + 252. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 - 54.2T + 6.89e4T^{2} \) |
| 43 | \( 1 + 438.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (128. - 222. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-515. + 297. i)T + (7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-238. - 412. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (548. + 316. i)T + (1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (308. + 534. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 396. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (-39.8 + 23.0i)T + (1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-344. + 597. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 - 1.28e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-595. + 1.03e3i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 - 946. iT - 9.12e5T^{2} \) |
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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
−10.39033255132223446307252035322, −9.200468679248120252225301759790, −8.714878341286096132585647049173, −7.76357544624202802657048234644, −6.55473936152923530843647407093, −5.57843283647840569725159065295, −4.63952294877597824794562305434, −3.60769440384979814680204573858, −1.79721921098084987977777653877, −0.877118747283001260014923348024,
1.48000004388989982794463336018, 2.48900529547235779030213199234, 3.93633887065611694886295832159, 4.99570188559390707912049876834, 6.22469041271998983149394408015, 6.89715411418316505449608894661, 7.954055098263576115564792818830, 8.996741126279819207814976611216, 9.831157784738598172376937195778, 10.64487105993006207910918284651