| L(s) = 1 | + (1.36 + 0.363i)2-s + (1.76 − 1.23i)3-s + (1.73 + 0.993i)4-s + (0.155 − 0.0136i)5-s + (2.85 − 1.04i)6-s + (−0.192 − 0.333i)7-s + (2.01 + 1.98i)8-s + (0.556 − 1.52i)9-s + (0.217 + 0.0379i)10-s + (−5.47 + 1.46i)11-s + (4.28 − 0.390i)12-s + (−4.75 − 3.33i)13-s + (−0.141 − 0.526i)14-s + (0.257 − 0.216i)15-s + (2.02 + 3.44i)16-s + (−1.56 + 0.569i)17-s + ⋯ |
| L(s) = 1 | + (0.966 + 0.257i)2-s + (1.01 − 0.712i)3-s + (0.867 + 0.496i)4-s + (0.0696 − 0.00609i)5-s + (1.16 − 0.426i)6-s + (−0.0728 − 0.126i)7-s + (0.710 + 0.703i)8-s + (0.185 − 0.509i)9-s + (0.0688 + 0.0120i)10-s + (−1.65 + 0.442i)11-s + (1.23 − 0.112i)12-s + (−1.31 − 0.923i)13-s + (−0.0379 − 0.140i)14-s + (0.0664 − 0.0557i)15-s + (0.506 + 0.862i)16-s + (−0.379 + 0.138i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0748i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0748i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.79408 - 0.104665i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.79408 - 0.104665i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.36 - 0.363i)T \) |
| 19 | \( 1 + (-1.19 - 4.19i)T \) |
| good | 3 | \( 1 + (-1.76 + 1.23i)T + (1.02 - 2.81i)T^{2} \) |
| 5 | \( 1 + (-0.155 + 0.0136i)T + (4.92 - 0.868i)T^{2} \) |
| 7 | \( 1 + (0.192 + 0.333i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (5.47 - 1.46i)T + (9.52 - 5.5i)T^{2} \) |
| 13 | \( 1 + (4.75 + 3.33i)T + (4.44 + 12.2i)T^{2} \) |
| 17 | \( 1 + (1.56 - 0.569i)T + (13.0 - 10.9i)T^{2} \) |
| 23 | \( 1 + (-6.91 + 5.80i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (-7.15 + 3.33i)T + (18.6 - 22.2i)T^{2} \) |
| 31 | \( 1 + (1.03 + 1.79i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.35 - 1.35i)T + 37iT^{2} \) |
| 41 | \( 1 + (-1.46 + 8.32i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-0.300 - 3.43i)T + (-42.3 + 7.46i)T^{2} \) |
| 47 | \( 1 + (2.21 - 6.09i)T + (-36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (-1.37 - 0.120i)T + (52.1 + 9.20i)T^{2} \) |
| 59 | \( 1 + (3.15 - 6.75i)T + (-37.9 - 45.1i)T^{2} \) |
| 61 | \( 1 + (5.87 + 0.514i)T + (60.0 + 10.5i)T^{2} \) |
| 67 | \( 1 + (-1.66 - 3.56i)T + (-43.0 + 51.3i)T^{2} \) |
| 71 | \( 1 + (-5.03 + 6.00i)T + (-12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (-5.43 - 0.958i)T + (68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (1.72 - 9.79i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (12.6 + 3.38i)T + (71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + (1.74 + 9.90i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (-0.789 - 2.16i)T + (-74.3 + 62.3i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.29756004680014538913528704621, −10.79731197603286760112535288097, −9.983247696239349283313857269730, −8.354346511158744945228046615542, −7.72614477940800589943661180977, −7.06957427483510727101485890140, −5.62324367489259980708857770218, −4.63567324687525028003935686438, −2.94692111970026921844792050373, −2.33494396707158500893439041116,
2.47216011414970882730006554555, 3.14481078634738532850522114775, 4.58153231475312333387359610147, 5.28482587369188692966930838051, 6.84224147669415478106263627689, 7.83468173367212866975710268842, 9.151270479807795830278173170188, 9.870131937581319001755710885052, 10.83958387664080173764293609188, 11.75914764599542456064824482822
