| L(s) = 1 | + (−0.424 + 1.86i)2-s + (0.113 − 0.142i)3-s + (3.92 + 1.89i)4-s + (11.1 − 13.9i)5-s + (0.216 + 0.271i)6-s + (1.18 + 18.4i)7-s + (−14.7 + 18.4i)8-s + (6.00 + 26.2i)9-s + (21.2 + 26.6i)10-s + (4.46 − 19.5i)11-s + (0.715 − 0.344i)12-s + (9.93 − 43.5i)13-s + (−34.8 − 5.64i)14-s + (−0.724 − 3.17i)15-s + (−6.32 − 7.93i)16-s + (9.17 − 4.41i)17-s + ⋯ |
| L(s) = 1 | + (−0.150 + 0.657i)2-s + (0.0218 − 0.0274i)3-s + (0.490 + 0.236i)4-s + (0.996 − 1.24i)5-s + (0.0147 + 0.0185i)6-s + (0.0638 + 0.997i)7-s + (−0.649 + 0.814i)8-s + (0.222 + 0.973i)9-s + (0.672 + 0.843i)10-s + (0.122 − 0.536i)11-s + (0.0172 − 0.00829i)12-s + (0.211 − 0.928i)13-s + (−0.666 − 0.107i)14-s + (−0.0124 − 0.0546i)15-s + (−0.0988 − 0.123i)16-s + (0.130 − 0.0630i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.672 - 0.739i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.672 - 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.46218 + 0.646879i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.46218 + 0.646879i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-1.18 - 18.4i)T \) |
| good | 2 | \( 1 + (0.424 - 1.86i)T + (-7.20 - 3.47i)T^{2} \) |
| 3 | \( 1 + (-0.113 + 0.142i)T + (-6.00 - 26.3i)T^{2} \) |
| 5 | \( 1 + (-11.1 + 13.9i)T + (-27.8 - 121. i)T^{2} \) |
| 11 | \( 1 + (-4.46 + 19.5i)T + (-1.19e3 - 577. i)T^{2} \) |
| 13 | \( 1 + (-9.93 + 43.5i)T + (-1.97e3 - 953. i)T^{2} \) |
| 17 | \( 1 + (-9.17 + 4.41i)T + (3.06e3 - 3.84e3i)T^{2} \) |
| 19 | \( 1 + 51.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + (140. + 67.4i)T + (7.58e3 + 9.51e3i)T^{2} \) |
| 29 | \( 1 + (-235. + 113. i)T + (1.52e4 - 1.90e4i)T^{2} \) |
| 31 | \( 1 + 211.T + 2.97e4T^{2} \) |
| 37 | \( 1 + (295. - 142. i)T + (3.15e4 - 3.96e4i)T^{2} \) |
| 41 | \( 1 + (-157. + 197. i)T + (-1.53e4 - 6.71e4i)T^{2} \) |
| 43 | \( 1 + (12.1 + 15.2i)T + (-1.76e4 + 7.75e4i)T^{2} \) |
| 47 | \( 1 + (1.76 - 7.74i)T + (-9.35e4 - 4.50e4i)T^{2} \) |
| 53 | \( 1 + (233. + 112. i)T + (9.28e4 + 1.16e5i)T^{2} \) |
| 59 | \( 1 + (-553. - 694. i)T + (-4.57e4 + 2.00e5i)T^{2} \) |
| 61 | \( 1 + (179. - 86.3i)T + (1.41e5 - 1.77e5i)T^{2} \) |
| 67 | \( 1 - 454.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (9.57 + 4.61i)T + (2.23e5 + 2.79e5i)T^{2} \) |
| 73 | \( 1 + (6.65 + 29.1i)T + (-3.50e5 + 1.68e5i)T^{2} \) |
| 79 | \( 1 - 906.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (26.1 + 114. i)T + (-5.15e5 + 2.48e5i)T^{2} \) |
| 89 | \( 1 + (46.7 + 204. i)T + (-6.35e5 + 3.05e5i)T^{2} \) |
| 97 | \( 1 - 1.20e3T + 9.12e5T^{2} \) |
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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
−15.65582592970162023371338014550, −14.16481964037653422058340067530, −12.95826415728211195149361949330, −12.02685495401322967779168221680, −10.39817776709362372975782620488, −8.783663300162059554198610469972, −8.112242223867212825243107925871, −6.11091877462552474909393691737, −5.23975809242278884872274015463, −2.20118251380174530192906452251,
1.82233001283256547054512356601, 3.61918136217418455283715231543, 6.35197600594681506502757905355, 6.99914563509665572776623623529, 9.522997823794389106424828970452, 10.26332250227834979094875208602, 11.16732831241614847644519021245, 12.46987635562461693925341357942, 14.02473428961935487931054416840, 14.68325016104020664549543283038
