L(s) = 1 | + (1.32 − 2.29i)2-s + (−2.52 − 4.37i)4-s + (0.5 − 0.866i)5-s + (−2.35 − 1.20i)7-s − 8.10·8-s + (−1.32 − 2.29i)10-s + (2.19 + 3.80i)11-s + 3.39·13-s + (−5.90 + 3.80i)14-s + (−5.70 + 9.88i)16-s + (−1.32 − 2.29i)17-s + (3.68 − 6.37i)19-s − 5.05·20-s + 11.6·22-s + (−2.37 + 4.12i)23-s + ⋯ |
L(s) = 1 | + (0.938 − 1.62i)2-s + (−1.26 − 2.18i)4-s + (0.223 − 0.387i)5-s + (−0.889 − 0.456i)7-s − 2.86·8-s + (−0.419 − 0.727i)10-s + (0.662 + 1.14i)11-s + 0.941·13-s + (−1.57 + 1.01i)14-s + (−1.42 + 2.47i)16-s + (−0.322 − 0.557i)17-s + (0.844 − 1.46i)19-s − 1.12·20-s + 2.48·22-s + (−0.496 + 0.859i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 + 0.112i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.993 + 0.112i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.102920 - 1.82877i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.102920 - 1.82877i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 + (2.35 + 1.20i)T \) |
good | 2 | \( 1 + (-1.32 + 2.29i)T + (-1 - 1.73i)T^{2} \) |
| 11 | \( 1 + (-2.19 - 3.80i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 3.39T + 13T^{2} \) |
| 17 | \( 1 + (1.32 + 2.29i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.68 + 6.37i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.37 - 4.12i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 3.70T + 29T^{2} \) |
| 31 | \( 1 + (0.759 + 1.31i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.223 + 0.387i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 6.49T + 41T^{2} \) |
| 43 | \( 1 + 1.13T + 43T^{2} \) |
| 47 | \( 1 + (1.25 - 2.18i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.65 - 4.59i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-7.30 - 12.6i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.12 + 7.15i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.95 - 6.85i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 12.2T + 71T^{2} \) |
| 73 | \( 1 + (7.53 + 13.0i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (1.28 - 2.22i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 0.552T + 83T^{2} \) |
| 89 | \( 1 + (6.45 - 11.1i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 2.53T + 97T^{2} \) |
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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
−11.45151117098413914420403398529, −10.43419269058973233059127095804, −9.556079810204737484070231998413, −9.108970403006991275767068430609, −7.07142531035242709320812885713, −5.89128405879752038034787870371, −4.69552996851458595905649911994, −3.85313628129206374508568373726, −2.64666726610654559022465758022, −1.11828419229209517751424452134,
3.23014103741360722219168224323, 3.99498216538527032029781735514, 5.69792338192610628999762320599, 6.10874810953832089903396636048, 6.87881231143537649571763388450, 8.236483045612908990622833654022, 8.776285498841957277693910787546, 10.04630284664599118162738626893, 11.51570896714280516471961787124, 12.50270226943557384005130744062