| L(s) = 1 | + (−0.192 − 0.716i)2-s + (1.25 − 0.724i)4-s + (−1.24 + 0.332i)7-s + (−1.80 − 1.80i)8-s + (−3.67 − 2.12i)11-s + (−5.52 − 1.47i)13-s + (0.476 + 0.825i)14-s + (0.500 − 0.866i)16-s + (−2.33 + 2.33i)17-s − 0.449i·19-s + (−0.814 + 3.04i)22-s + (1.90 − 7.09i)23-s + 4.24i·26-s + (−1.31 + 1.31i)28-s + (2.59 − 4.5i)29-s + ⋯ |
| L(s) = 1 | + (−0.135 − 0.506i)2-s + (0.627 − 0.362i)4-s + (−0.469 + 0.125i)7-s + (−0.639 − 0.639i)8-s + (−1.10 − 0.639i)11-s + (−1.53 − 0.410i)13-s + (0.127 + 0.220i)14-s + (0.125 − 0.216i)16-s + (−0.566 + 0.566i)17-s − 0.103i·19-s + (−0.173 + 0.648i)22-s + (0.396 − 1.47i)23-s + 0.832i·26-s + (−0.248 + 0.248i)28-s + (0.482 − 0.835i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.962 + 0.269i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.962 + 0.269i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.116257 - 0.845934i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.116257 - 0.845934i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + (0.192 + 0.716i)T + (-1.73 + i)T^{2} \) |
| 7 | \( 1 + (1.24 - 0.332i)T + (6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + (3.67 + 2.12i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (5.52 + 1.47i)T + (11.2 + 6.5i)T^{2} \) |
| 17 | \( 1 + (2.33 - 2.33i)T - 17iT^{2} \) |
| 19 | \( 1 + 0.449iT - 19T^{2} \) |
| 23 | \( 1 + (-1.90 + 7.09i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + (-2.59 + 4.5i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (0.224 + 0.389i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-5.86 - 5.86i)T + 37iT^{2} \) |
| 41 | \( 1 + (-8.17 + 4.71i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (1.47 + 5.52i)T + (-37.2 + 21.5i)T^{2} \) |
| 47 | \( 1 + (-1.04 - 3.90i)T + (-40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (3.61 + 3.61i)T + 53iT^{2} \) |
| 59 | \( 1 + (-4.24 - 7.34i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (1.27 - 2.20i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.81 + 6.76i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 - 6.14iT - 71T^{2} \) |
| 73 | \( 1 - 73iT^{2} \) |
| 79 | \( 1 + (6.75 + 3.89i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.22 + 1.13i)T + (71.8 - 41.5i)T^{2} \) |
| 89 | \( 1 + 13.6T + 89T^{2} \) |
| 97 | \( 1 + (-8.56 + 2.29i)T + (84.0 - 48.5i)T^{2} \) |
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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
−10.25943737986463877851316643275, −9.528635882677633352608051422417, −8.416786311668656574250605398397, −7.45608147697764899571990424828, −6.48813032918146638363321422475, −5.67532939043876004470962139433, −4.53593243844290430051888583726, −2.91484465147929279884433175969, −2.39768188845846978683421772713, −0.42024099438633301546394545833,
2.22472699700927129964918731994, 3.08410836216874293349019274935, 4.65203024954152531304028028497, 5.55347209701253461942605385624, 6.75381364900623919716220615343, 7.35680593215730241422830949506, 7.958961387451885355228807621187, 9.262518773208136911875152066196, 9.850422111702320215502589110413, 10.95263559525499677363161207027
