| L(s) = 1 | + (0.151 + 0.860i)2-s + (0.910 + 1.47i)3-s + (1.16 − 0.423i)4-s + (−0.766 − 0.642i)5-s + (−1.12 + 1.00i)6-s + (−0.939 − 0.342i)7-s + (1.41 + 2.44i)8-s + (−1.34 + 2.68i)9-s + (0.436 − 0.756i)10-s + (2.44 − 2.04i)11-s + (1.68 + 1.32i)12-s + (−0.613 + 3.48i)13-s + (0.151 − 0.860i)14-s + (0.250 − 1.71i)15-s + (0.00335 − 0.00281i)16-s + (−0.965 + 1.67i)17-s + ⋯ |
| L(s) = 1 | + (0.107 + 0.608i)2-s + (0.525 + 0.850i)3-s + (0.581 − 0.211i)4-s + (−0.342 − 0.287i)5-s + (−0.461 + 0.410i)6-s + (−0.355 − 0.129i)7-s + (0.499 + 0.865i)8-s + (−0.447 + 0.894i)9-s + (0.138 − 0.239i)10-s + (0.735 − 0.617i)11-s + (0.485 + 0.383i)12-s + (−0.170 + 0.965i)13-s + (0.0405 − 0.229i)14-s + (0.0645 − 0.442i)15-s + (0.000839 − 0.000704i)16-s + (−0.234 + 0.405i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.315 - 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.315 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.29784 + 1.80009i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.29784 + 1.80009i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.910 - 1.47i)T \) |
| 5 | \( 1 + (0.766 + 0.642i)T \) |
| 7 | \( 1 + (0.939 + 0.342i)T \) |
| good | 2 | \( 1 + (-0.151 - 0.860i)T + (-1.87 + 0.684i)T^{2} \) |
| 11 | \( 1 + (-2.44 + 2.04i)T + (1.91 - 10.8i)T^{2} \) |
| 13 | \( 1 + (0.613 - 3.48i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (0.965 - 1.67i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.69 - 4.67i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-5.92 + 2.15i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (-0.203 - 1.15i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (8.56 - 3.11i)T + (23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (3.32 - 5.76i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.43 + 8.16i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-4.18 + 3.51i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (-11.1 - 4.04i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + 7.07T + 53T^{2} \) |
| 59 | \( 1 + (2.09 + 1.75i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-2.36 - 0.860i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-0.338 + 1.91i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-5.46 + 9.46i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-2.52 - 4.37i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (2.83 + 16.0i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (1.48 + 8.42i)T + (-77.9 + 28.3i)T^{2} \) |
| 89 | \( 1 + (7.20 + 12.4i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (0.974 - 0.817i)T + (16.8 - 95.5i)T^{2} \) |
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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
−10.38131844547608615262968937997, −9.158560303385838035557654481912, −8.812153386892943247793688309720, −7.70448633450840062165336586912, −6.96358929057885056847878251761, −5.97230209589372714248608703194, −5.09790737130907619305390340496, −4.05760094406656251413591245015, −3.17395563063108832606539831479, −1.72813205575467865450949040718,
0.994483664305262331927773243536, 2.38593812234005989557116359499, 3.07333837639783583108587804244, 3.99521457589837688858415040807, 5.53958731989916958512973526754, 6.80622694194705167418933788894, 7.15276556304758134027689206751, 7.889951384956364764696223150857, 9.145419330839833715745344416503, 9.659573092560119693440886787452
