| L(s) = 1 | + (−4.51 + 0.795i)2-s + (−1.14 + 6.46i)3-s + (12.2 − 4.44i)4-s + (−2.86 + 3.40i)5-s − 30.0i·6-s + (−25.5 − 21.4i)7-s + (−19.7 + 11.4i)8-s + (−15.1 − 5.52i)9-s + (10.1 − 17.6i)10-s + (5.32 + 9.22i)11-s + (14.8 + 84.0i)12-s + (−18.5 − 51.0i)13-s + (132. + 76.3i)14-s + (−18.7 − 22.3i)15-s + (0.571 − 0.479i)16-s + (−36.5 + 100. i)17-s + ⋯ |
| L(s) = 1 | + (−1.59 + 0.281i)2-s + (−0.219 + 1.24i)3-s + (1.52 − 0.555i)4-s + (−0.255 + 0.304i)5-s − 2.04i·6-s + (−1.37 − 1.15i)7-s + (−0.874 + 0.504i)8-s + (−0.562 − 0.204i)9-s + (0.322 − 0.558i)10-s + (0.146 + 0.252i)11-s + (0.356 + 2.02i)12-s + (−0.396 − 1.08i)13-s + (2.52 + 1.45i)14-s + (−0.323 − 0.385i)15-s + (0.00893 − 0.00749i)16-s + (−0.521 + 1.43i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.699 + 0.714i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.699 + 0.714i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0316098 - 0.0751210i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0316098 - 0.0751210i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + (-42.9 - 220. i)T \) |
| good | 2 | \( 1 + (4.51 - 0.795i)T + (7.51 - 2.73i)T^{2} \) |
| 3 | \( 1 + (1.14 - 6.46i)T + (-25.3 - 9.23i)T^{2} \) |
| 5 | \( 1 + (2.86 - 3.40i)T + (-21.7 - 123. i)T^{2} \) |
| 7 | \( 1 + (25.5 + 21.4i)T + (59.5 + 337. i)T^{2} \) |
| 11 | \( 1 + (-5.32 - 9.22i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (18.5 + 51.0i)T + (-1.68e3 + 1.41e3i)T^{2} \) |
| 17 | \( 1 + (36.5 - 100. i)T + (-3.76e3 - 3.15e3i)T^{2} \) |
| 19 | \( 1 + (4.24 + 0.748i)T + (6.44e3 + 2.34e3i)T^{2} \) |
| 23 | \( 1 + (163. + 94.4i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (97.9 - 56.5i)T + (1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + 126. iT - 2.97e4T^{2} \) |
| 41 | \( 1 + (-45.8 + 16.7i)T + (5.27e4 - 4.43e4i)T^{2} \) |
| 43 | \( 1 + 137. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (162. - 282. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-161. + 135. i)T + (2.58e4 - 1.46e5i)T^{2} \) |
| 59 | \( 1 + (216. + 258. i)T + (-3.56e4 + 2.02e5i)T^{2} \) |
| 61 | \( 1 + (135. + 373. i)T + (-1.73e5 + 1.45e5i)T^{2} \) |
| 67 | \( 1 + (170. + 142. i)T + (5.22e4 + 2.96e5i)T^{2} \) |
| 71 | \( 1 + (116. - 663. i)T + (-3.36e5 - 1.22e5i)T^{2} \) |
| 73 | \( 1 - 782.T + 3.89e5T^{2} \) |
| 79 | \( 1 + (92.5 - 110. i)T + (-8.56e4 - 4.85e5i)T^{2} \) |
| 83 | \( 1 + (-648. - 236. i)T + (4.38e5 + 3.67e5i)T^{2} \) |
| 89 | \( 1 + (-214. - 255. i)T + (-1.22e5 + 6.94e5i)T^{2} \) |
| 97 | \( 1 + (-1.45e3 - 839. i)T + (4.56e5 + 7.90e5i)T^{2} \) |
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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
−16.73507541057090476667126813680, −15.85105109555994940250189187084, −15.02449838861208624760137967344, −12.96620895784031055340488241300, −10.80319312342052955000637342931, −10.25426168426430827307785123683, −9.487882784069036099091071315019, −7.83892116255409626703768797833, −6.45626198746264947455374892652, −3.86575959281851778409195788685,
0.11189597654072463919870728179, 2.24327104321897911679282505082, 6.33236451053227096531378547245, 7.42217043160628900030601621526, 8.888010834257443289522066929737, 9.697644801901662505027552143373, 11.70098028890655764215037763838, 12.20306596659434844092208614631, 13.63517249840481819541582800920, 15.88189515184282811260332333729
