L(s) = 1 | + (−1.53 + 0.795i)3-s − 3.80·5-s + (1.73 − 2.44i)9-s − 0.357i·11-s − 4.04i·13-s + (5.84 − 3.02i)15-s − 0.103·17-s − 2.45i·19-s − 1.33i·23-s + 9.44·25-s + (−0.723 + 5.14i)27-s + 4.97i·29-s + 7.88i·31-s + (0.284 + 0.549i)33-s − 10.9·37-s + ⋯ |
L(s) = 1 | + (−0.888 + 0.459i)3-s − 1.69·5-s + (0.578 − 0.815i)9-s − 0.107i·11-s − 1.12i·13-s + (1.50 − 0.780i)15-s − 0.0252·17-s − 0.563i·19-s − 0.277i·23-s + 1.88·25-s + (−0.139 + 0.990i)27-s + 0.923i·29-s + 1.41i·31-s + (0.0494 + 0.0957i)33-s − 1.79·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.234 - 0.972i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.234 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5334652400\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5334652400\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.53 - 0.795i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 3.80T + 5T^{2} \) |
| 11 | \( 1 + 0.357iT - 11T^{2} \) |
| 13 | \( 1 + 4.04iT - 13T^{2} \) |
| 17 | \( 1 + 0.103T + 17T^{2} \) |
| 19 | \( 1 + 2.45iT - 19T^{2} \) |
| 23 | \( 1 + 1.33iT - 23T^{2} \) |
| 29 | \( 1 - 4.97iT - 29T^{2} \) |
| 31 | \( 1 - 7.88iT - 31T^{2} \) |
| 37 | \( 1 + 10.9T + 37T^{2} \) |
| 41 | \( 1 + 6.15T + 41T^{2} \) |
| 43 | \( 1 - 0.502T + 43T^{2} \) |
| 47 | \( 1 - 11.4T + 47T^{2} \) |
| 53 | \( 1 - 5.86iT - 53T^{2} \) |
| 59 | \( 1 - 7.54T + 59T^{2} \) |
| 61 | \( 1 - 9.47iT - 61T^{2} \) |
| 67 | \( 1 - 2.68T + 67T^{2} \) |
| 71 | \( 1 - 5.78iT - 71T^{2} \) |
| 73 | \( 1 - 0.235iT - 73T^{2} \) |
| 79 | \( 1 - 3.22T + 79T^{2} \) |
| 83 | \( 1 - 9.07T + 83T^{2} \) |
| 89 | \( 1 - 6.82T + 89T^{2} \) |
| 97 | \( 1 - 5.14iT - 97T^{2} \) |
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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
−10.38365396039334507351493856000, −8.968543080298670315203541303728, −8.381768232009046668144741683567, −7.30096901790950540232942038062, −6.82412948934510176758551616594, −5.48932534430074784209912338247, −4.83168960219882486351955199355, −3.84712076151549953996996747266, −3.13734202847238576034699645626, −0.842845732733168702852969916292,
0.38276390526987725054237027754, 1.97773094138637923515548758349, 3.68495880148268561846731597474, 4.32633009926783421982444519469, 5.28769947634455895356019715424, 6.43080806974897091056334535216, 7.15022083968076295710476479544, 7.80353682066296652957357851804, 8.540152567538523164316044919651, 9.678147806445802513028883658993