L(s) = 1 | + (−0.707 + 0.707i)2-s + (0.668 − 1.59i)3-s − 1.00i·4-s + (−0.595 + 2.15i)5-s + (0.656 + 1.60i)6-s + (−0.0702 − 0.0702i)7-s + (0.707 + 0.707i)8-s + (−2.10 − 2.13i)9-s + (−1.10 − 1.94i)10-s − 3.25i·11-s + (−1.59 − 0.668i)12-s + (1.84 − 1.84i)13-s + 0.0993·14-s + (3.04 + 2.39i)15-s − 1.00·16-s + (3.76 − 3.76i)17-s + ⋯ |
L(s) = 1 | + (−0.499 + 0.499i)2-s + (0.386 − 0.922i)3-s − 0.500i·4-s + (−0.266 + 0.963i)5-s + (0.268 + 0.654i)6-s + (−0.0265 − 0.0265i)7-s + (0.250 + 0.250i)8-s + (−0.701 − 0.712i)9-s + (−0.348 − 0.615i)10-s − 0.981i·11-s + (−0.461 − 0.193i)12-s + (0.512 − 0.512i)13-s + 0.0265·14-s + (0.786 + 0.617i)15-s − 0.250·16-s + (0.913 − 0.913i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.623 + 0.782i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.623 + 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.03401 - 0.498163i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.03401 - 0.498163i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 - 0.707i)T \) |
| 3 | \( 1 + (-0.668 + 1.59i)T \) |
| 5 | \( 1 + (0.595 - 2.15i)T \) |
| 19 | \( 1 - iT \) |
good | 7 | \( 1 + (0.0702 + 0.0702i)T + 7iT^{2} \) |
| 11 | \( 1 + 3.25iT - 11T^{2} \) |
| 13 | \( 1 + (-1.84 + 1.84i)T - 13iT^{2} \) |
| 17 | \( 1 + (-3.76 + 3.76i)T - 17iT^{2} \) |
| 23 | \( 1 + (1.02 + 1.02i)T + 23iT^{2} \) |
| 29 | \( 1 - 6.98T + 29T^{2} \) |
| 31 | \( 1 - 9.93T + 31T^{2} \) |
| 37 | \( 1 + (3.00 + 3.00i)T + 37iT^{2} \) |
| 41 | \( 1 + 8.72iT - 41T^{2} \) |
| 43 | \( 1 + (2.35 - 2.35i)T - 43iT^{2} \) |
| 47 | \( 1 + (0.871 - 0.871i)T - 47iT^{2} \) |
| 53 | \( 1 + (8.49 + 8.49i)T + 53iT^{2} \) |
| 59 | \( 1 + 2.24T + 59T^{2} \) |
| 61 | \( 1 - 8.33T + 61T^{2} \) |
| 67 | \( 1 + (-2.25 - 2.25i)T + 67iT^{2} \) |
| 71 | \( 1 - 6.67iT - 71T^{2} \) |
| 73 | \( 1 + (4.70 - 4.70i)T - 73iT^{2} \) |
| 79 | \( 1 - 9.81iT - 79T^{2} \) |
| 83 | \( 1 + (-10.5 - 10.5i)T + 83iT^{2} \) |
| 89 | \( 1 - 0.330T + 89T^{2} \) |
| 97 | \( 1 + (3.68 + 3.68i)T + 97iT^{2} \) |
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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
−10.52615441956288673529562576315, −9.699481881725595018446928858999, −8.365029218400800111038187138053, −8.110160044020078564874088452770, −7.00464282779662895712235473751, −6.40110647885218175733652541301, −5.46069855849191027400868015158, −3.54463939767197420616814156037, −2.61831316287857936256643059958, −0.807001809801986964309268410984,
1.52645091191872966534021736365, 3.07452323328522801986324151335, 4.25207025998226902728648043303, 4.84829977311111195303754061628, 6.27702621544641847176973259299, 7.84131858777189515670013612996, 8.356152969761129564205027656990, 9.244480756232154695904681606565, 9.913544545110400385226260922981, 10.60160764053340362985914177577