L(s) = 1 | + (0.5 + 0.866i)3-s + (0.5 − 0.866i)4-s + (−0.499 + 0.866i)9-s + 0.999·12-s + (−0.499 − 0.866i)16-s − 25-s − 0.999·27-s + (0.499 + 0.866i)36-s + (−1 + 1.73i)43-s + (0.499 − 0.866i)48-s + (−0.5 − 0.866i)49-s + (1 − 1.73i)61-s − 0.999·64-s + (−0.5 − 0.866i)75-s − 2·79-s + ⋯ |
L(s) = 1 | + (0.5 + 0.866i)3-s + (0.5 − 0.866i)4-s + (−0.499 + 0.866i)9-s + 0.999·12-s + (−0.499 − 0.866i)16-s − 25-s − 0.999·27-s + (0.499 + 0.866i)36-s + (−1 + 1.73i)43-s + (0.499 − 0.866i)48-s + (−0.5 − 0.866i)49-s + (1 − 1.73i)61-s − 0.999·64-s + (−0.5 − 0.866i)75-s − 2·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.089558674\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.089558674\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 5 | \( 1 + T^{2} \) |
| 7 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + 2T + T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (0.5 + 0.866i)T^{2} \) |
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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
−11.11886424873715407517357760874, −10.04713579963038427843560085613, −9.715119311473587752385605383509, −8.619170204653682386944463403602, −7.64778581848165550263809601842, −6.45163948241996375246769418140, −5.46928895015017093649496777937, −4.55170743068002638995567224754, −3.25510376772767607229686246654, −1.97836346568506537680213813821,
1.90836237988088961761687891822, 3.01664450915322485633931603272, 4.04570128725164222587001565108, 5.75358915284655606257774059667, 6.78226750023478199095015630063, 7.47486373456697764431656940157, 8.282063256949368370755606003582, 9.011768506709543287781317351895, 10.20845153979420460309348321019, 11.44895139159288995580079601776