| L(s) = 1 | + (1.66 + 2.56i)2-s + (3.28 + 8.56i)3-s + (−0.554 + 1.24i)4-s + (5.95 − 9.46i)5-s + (−16.4 + 22.7i)6-s + (16.8 + 7.73i)7-s + (20.0 − 3.17i)8-s + (−42.4 + 38.2i)9-s + (34.2 − 0.487i)10-s + (−18.2 + 20.2i)11-s + (−12.4 − 0.654i)12-s + (−30.7 + 15.6i)13-s + (8.18 + 56.0i)14-s + (100. + 19.8i)15-s + (48.8 + 54.2i)16-s + (−33.3 − 41.1i)17-s + ⋯ |
| L(s) = 1 | + (0.589 + 0.907i)2-s + (0.632 + 1.64i)3-s + (−0.0693 + 0.155i)4-s + (0.532 − 0.846i)5-s + (−1.12 + 1.54i)6-s + (0.908 + 0.417i)7-s + (0.886 − 0.140i)8-s + (−1.57 + 1.41i)9-s + (1.08 − 0.0154i)10-s + (−0.500 + 0.556i)11-s + (−0.300 − 0.0157i)12-s + (−0.656 + 0.334i)13-s + (0.156 + 1.07i)14-s + (1.73 + 0.342i)15-s + (0.763 + 0.848i)16-s + (−0.475 − 0.587i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.633 - 0.774i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.633 - 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.43296 + 3.02304i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.43296 + 3.02304i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-5.95 + 9.46i)T \) |
| 7 | \( 1 + (-16.8 - 7.73i)T \) |
| good | 2 | \( 1 + (-1.66 - 2.56i)T + (-3.25 + 7.30i)T^{2} \) |
| 3 | \( 1 + (-3.28 - 8.56i)T + (-20.0 + 18.0i)T^{2} \) |
| 11 | \( 1 + (18.2 - 20.2i)T + (-139. - 1.32e3i)T^{2} \) |
| 13 | \( 1 + (30.7 - 15.6i)T + (1.29e3 - 1.77e3i)T^{2} \) |
| 17 | \( 1 + (33.3 + 41.1i)T + (-1.02e3 + 4.80e3i)T^{2} \) |
| 19 | \( 1 + (-17.3 + 7.74i)T + (4.58e3 - 5.09e3i)T^{2} \) |
| 23 | \( 1 + (-91.8 + 59.6i)T + (4.94e3 - 1.11e4i)T^{2} \) |
| 29 | \( 1 + (142. + 195. i)T + (-7.53e3 + 2.31e4i)T^{2} \) |
| 31 | \( 1 + (-112. - 11.8i)T + (2.91e4 + 6.19e3i)T^{2} \) |
| 37 | \( 1 + (-9.15 + 174. i)T + (-5.03e4 - 5.29e3i)T^{2} \) |
| 41 | \( 1 + (190. + 62.0i)T + (5.57e4 + 4.05e4i)T^{2} \) |
| 43 | \( 1 + (131. - 131. i)T - 7.95e4iT^{2} \) |
| 47 | \( 1 + (-440. - 356. i)T + (2.15e4 + 1.01e5i)T^{2} \) |
| 53 | \( 1 + (275. - 105. i)T + (1.10e5 - 9.96e4i)T^{2} \) |
| 59 | \( 1 + (-425. + 90.3i)T + (1.87e5 - 8.35e4i)T^{2} \) |
| 61 | \( 1 + (-14.1 + 66.3i)T + (-2.07e5 - 9.23e4i)T^{2} \) |
| 67 | \( 1 + (-594. + 481. i)T + (6.25e4 - 2.94e5i)T^{2} \) |
| 71 | \( 1 + (785. - 570. i)T + (1.10e5 - 3.40e5i)T^{2} \) |
| 73 | \( 1 + (540. - 28.3i)T + (3.86e5 - 4.06e4i)T^{2} \) |
| 79 | \( 1 + (708. - 74.4i)T + (4.82e5 - 1.02e5i)T^{2} \) |
| 83 | \( 1 + (185. + 1.17e3i)T + (-5.43e5 + 1.76e5i)T^{2} \) |
| 89 | \( 1 + (-819. - 174. i)T + (6.44e5 + 2.86e5i)T^{2} \) |
| 97 | \( 1 + (-223. + 1.40e3i)T + (-8.68e5 - 2.82e5i)T^{2} \) |
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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
−13.00411985694400650001629940073, −11.48390925648834767798740382903, −10.35627630417473940903309990007, −9.509874812199559279446020964226, −8.643155865632707473246481052059, −7.53719216110327275684907093953, −5.71945161274789665050283561223, −4.80665254007988519792913662249, −4.47169788620798254022033094383, −2.30251504759891671715868600132,
1.41582977540055375978419728647, 2.40293800716337947930207709889, 3.38382282065750220021560948472, 5.35933817903243504753054601367, 6.92300486557307239581889292585, 7.57271851640837300853678353569, 8.553973364580430735564072155702, 10.31794993057267973087860459602, 11.20935031660512147238079986159, 11.99126934667211952927147651536
