L(s) = 1 | + (0.607 + 2.26i)2-s + (−3.03 + 1.75i)4-s + (−1.12 − 1.12i)5-s + (−0.437 − 2.60i)7-s + (−2.50 − 2.50i)8-s + (1.87 − 3.23i)10-s + (3.03 − 0.813i)11-s + (−1.04 − 3.44i)13-s + (5.64 − 2.57i)14-s + (0.649 − 1.12i)16-s + (−0.320 − 0.555i)17-s + (2.04 − 7.61i)19-s + (5.40 + 1.44i)20-s + (3.68 + 6.38i)22-s + (0.126 + 0.0730i)23-s + ⋯ |
L(s) = 1 | + (0.429 + 1.60i)2-s + (−1.51 + 0.877i)4-s + (−0.503 − 0.503i)5-s + (−0.165 − 0.986i)7-s + (−0.886 − 0.886i)8-s + (0.591 − 1.02i)10-s + (0.914 − 0.245i)11-s + (−0.290 − 0.956i)13-s + (1.51 − 0.689i)14-s + (0.162 − 0.281i)16-s + (−0.0778 − 0.134i)17-s + (0.468 − 1.74i)19-s + (1.20 + 0.323i)20-s + (0.786 + 1.36i)22-s + (0.0263 + 0.0152i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.911 - 0.410i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.911 - 0.410i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.42197 + 0.305095i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.42197 + 0.305095i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (0.437 + 2.60i)T \) |
| 13 | \( 1 + (1.04 + 3.44i)T \) |
good | 2 | \( 1 + (-0.607 - 2.26i)T + (-1.73 + i)T^{2} \) |
| 5 | \( 1 + (1.12 + 1.12i)T + 5iT^{2} \) |
| 11 | \( 1 + (-3.03 + 0.813i)T + (9.52 - 5.5i)T^{2} \) |
| 17 | \( 1 + (0.320 + 0.555i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.04 + 7.61i)T + (-16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (-0.126 - 0.0730i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.49 - 2.58i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-4.73 - 4.73i)T + 31iT^{2} \) |
| 37 | \( 1 + (-3.75 + 1.00i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (5.60 - 1.50i)T + (35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (-2.42 + 1.40i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-2.22 + 2.22i)T - 47iT^{2} \) |
| 53 | \( 1 - 7.32T + 53T^{2} \) |
| 59 | \( 1 + (-4.00 - 1.07i)T + (51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + (3.90 - 2.25i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (0.366 + 1.36i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (13.8 + 3.70i)T + (61.4 + 35.5i)T^{2} \) |
| 73 | \( 1 + (-4.99 + 4.99i)T - 73iT^{2} \) |
| 79 | \( 1 + 0.632T + 79T^{2} \) |
| 83 | \( 1 + (1.07 + 1.07i)T + 83iT^{2} \) |
| 89 | \( 1 + (3.51 + 13.1i)T + (-77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (-0.0487 + 0.181i)T + (-84.0 - 48.5i)T^{2} \) |
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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
−10.12702115984794336130362194478, −8.998246706943228440006598180499, −8.419868277597270483785134603490, −7.40647567262252157431168412164, −6.99110548154355472857756310803, −6.03439550499270085408077514151, −4.92108540834122433807843899411, −4.36936081083809468550157067738, −3.25745824158183836986921896080, −0.67358601879956537550854915262,
1.55802174299259660062961717389, 2.56024196500629057444244613521, 3.64847247792282641210715150735, 4.27922156437277206035192198925, 5.53567544448509044971076204147, 6.54913762961154592244319884794, 7.73023169628082435575083977559, 8.908165086246401094515619991533, 9.614797869644337365468588809482, 10.23042890394615094365803876392