| L(s) = 1 | + (−2.70 − 0.141i)2-s + (−2.31 − 2.85i)3-s + (−0.643 − 0.0676i)4-s + (8.55 + 7.20i)5-s + (5.85 + 8.05i)6-s + (−16.6 − 8.09i)7-s + (23.1 + 3.66i)8-s + (2.80 − 13.2i)9-s + (−22.1 − 20.7i)10-s + (30.3 − 6.44i)11-s + (1.29 + 1.99i)12-s + (−54.8 − 27.9i)13-s + (43.9 + 24.2i)14-s + (0.790 − 41.0i)15-s + (−57.1 − 12.1i)16-s + (20.8 + 54.2i)17-s + ⋯ |
| L(s) = 1 | + (−0.957 − 0.0501i)2-s + (−0.444 − 0.549i)3-s + (−0.0804 − 0.00845i)4-s + (0.764 + 0.644i)5-s + (0.398 + 0.548i)6-s + (−0.899 − 0.436i)7-s + (1.02 + 0.162i)8-s + (0.104 − 0.489i)9-s + (−0.699 − 0.655i)10-s + (0.831 − 0.176i)11-s + (0.0311 + 0.0479i)12-s + (−1.17 − 0.596i)13-s + (0.839 + 0.463i)14-s + (0.0136 − 0.706i)15-s + (−0.892 − 0.189i)16-s + (0.296 + 0.773i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.777 - 0.628i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.777 - 0.628i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.00673058 + 0.0190194i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00673058 + 0.0190194i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-8.55 - 7.20i)T \) |
| 7 | \( 1 + (16.6 + 8.09i)T \) |
| good | 2 | \( 1 + (2.70 + 0.141i)T + (7.95 + 0.836i)T^{2} \) |
| 3 | \( 1 + (2.31 + 2.85i)T + (-5.61 + 26.4i)T^{2} \) |
| 11 | \( 1 + (-30.3 + 6.44i)T + (1.21e3 - 541. i)T^{2} \) |
| 13 | \( 1 + (54.8 + 27.9i)T + (1.29e3 + 1.77e3i)T^{2} \) |
| 17 | \( 1 + (-20.8 - 54.2i)T + (-3.65e3 + 3.28e3i)T^{2} \) |
| 19 | \( 1 + (12.2 + 116. i)T + (-6.70e3 + 1.42e3i)T^{2} \) |
| 23 | \( 1 + (7.28 - 138. i)T + (-1.21e4 - 1.27e3i)T^{2} \) |
| 29 | \( 1 + (38.9 - 53.6i)T + (-7.53e3 - 2.31e4i)T^{2} \) |
| 31 | \( 1 + (95.5 - 214. i)T + (-1.99e4 - 2.21e4i)T^{2} \) |
| 37 | \( 1 + (158. - 103. i)T + (2.06e4 - 4.62e4i)T^{2} \) |
| 41 | \( 1 + (184. - 59.9i)T + (5.57e4 - 4.05e4i)T^{2} \) |
| 43 | \( 1 + (361. + 361. i)T + 7.95e4iT^{2} \) |
| 47 | \( 1 + (142. + 54.7i)T + (7.71e4 + 6.94e4i)T^{2} \) |
| 53 | \( 1 + (139. - 112. i)T + (3.09e4 - 1.45e5i)T^{2} \) |
| 59 | \( 1 + (-224. + 249. i)T + (-2.14e4 - 2.04e5i)T^{2} \) |
| 61 | \( 1 + (-21.6 + 19.5i)T + (2.37e4 - 2.25e5i)T^{2} \) |
| 67 | \( 1 + (152. - 58.4i)T + (2.23e5 - 2.01e5i)T^{2} \) |
| 71 | \( 1 + (-814. - 591. i)T + (1.10e5 + 3.40e5i)T^{2} \) |
| 73 | \( 1 + (662. - 1.01e3i)T + (-1.58e5 - 3.55e5i)T^{2} \) |
| 79 | \( 1 + (360. + 809. i)T + (-3.29e5 + 3.66e5i)T^{2} \) |
| 83 | \( 1 + (-24.2 + 153. i)T + (-5.43e5 - 1.76e5i)T^{2} \) |
| 89 | \( 1 + (453. + 503. i)T + (-7.36e4 + 7.01e5i)T^{2} \) |
| 97 | \( 1 + (145. + 916. i)T + (-8.68e5 + 2.82e5i)T^{2} \) |
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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
−12.71895256818765134356799428674, −11.46974739073954839932237734567, −10.30917702547196467770617926281, −9.731014630532988009718517972023, −8.850794880053864020027016921976, −7.18346340042681824122507371054, −6.77199363139231921984281112205, −5.36663646703510766931838317236, −3.42254829422624501170088409228, −1.47829721467928351451031374250,
0.01380748917246478427193977632, 1.95497545962538546312747069467, 4.26624467567141713751358582238, 5.30350936337814507183453940207, 6.60494378340039049781243454909, 7.992962401798487147227257896207, 9.225318359218388210917431190418, 9.714976819134299218087710272177, 10.32387182807938979207963750959, 11.83127021041590355882499407270
