L(s) = 1 | + (−1 + 1.73i)3-s + (−0.499 − 0.866i)9-s − 4·13-s + (−3 + 5.19i)17-s + (1 + 1.73i)19-s + (2.5 − 4.33i)25-s − 4.00·27-s − 6·29-s + (−2 + 3.46i)31-s + (−1 − 1.73i)37-s + (4 − 6.92i)39-s + 6·41-s − 8·43-s + (−6 − 10.3i)47-s + (−6 − 10.3i)51-s + ⋯ |
L(s) = 1 | + (−0.577 + 0.999i)3-s + (−0.166 − 0.288i)9-s − 1.10·13-s + (−0.727 + 1.26i)17-s + (0.229 + 0.397i)19-s + (0.5 − 0.866i)25-s − 0.769·27-s − 1.11·29-s + (−0.359 + 0.622i)31-s + (−0.164 − 0.284i)37-s + (0.640 − 1.10i)39-s + 0.937·41-s − 1.21·43-s + (−0.875 − 1.51i)47-s + (−0.840 − 1.45i)51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0317290 - 0.499982i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0317290 - 0.499982i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (1 - 1.73i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 17 | \( 1 + (3 - 5.19i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1 - 1.73i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + (2 - 3.46i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1 + 1.73i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 + (6 + 10.3i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (3 - 5.19i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (3 - 5.19i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2 - 3.46i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + (1 - 1.73i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-4 - 6.92i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 + (-3 - 5.19i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−10.61662441845157271609312247549, −10.04889135363310651345323215390, −9.244186465240802806910759944490, −8.268691822816116356781575522813, −7.25508198704575697242647246701, −6.20316109683639666590248107495, −5.25573868168774993405313395900, −4.50458714447059805024772520608, −3.57393654806864435044477955707, −2.03644058740327689306834047086,
0.25674874227866546480281024802, 1.79666454082796095718830451387, 3.03705038180725135820462610431, 4.60141690972574425880329641702, 5.44902148242624285052927347525, 6.51790002346906792265498944020, 7.22149221601780025571692144702, 7.77589947345195472371491411327, 9.192650234681562265281981519467, 9.659057556386878572095815492825