L(s) = 1 | + (−0.5 + 0.866i)5-s + (−0.5 − 0.866i)7-s + (0.5 + 0.866i)11-s + (1 + 1.73i)29-s + (−0.5 + 0.866i)31-s + 0.999·35-s + 53-s − 0.999·55-s + (−1 + 1.73i)59-s − 73-s + (0.499 − 0.866i)77-s + (1 + 1.73i)79-s + (0.5 + 0.866i)83-s + (0.5 + 0.866i)97-s + (−0.5 − 0.866i)101-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.866i)5-s + (−0.5 − 0.866i)7-s + (0.5 + 0.866i)11-s + (1 + 1.73i)29-s + (−0.5 + 0.866i)31-s + 0.999·35-s + 53-s − 0.999·55-s + (−1 + 1.73i)59-s − 73-s + (0.499 − 0.866i)77-s + (1 + 1.73i)79-s + (0.5 + 0.866i)83-s + (0.5 + 0.866i)97-s + (−0.5 − 0.866i)101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9666024653\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9666024653\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 7 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 - T + T^{2} \) |
| 59 | \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
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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
−9.236351314620202215975435214645, −8.462242980186701268989682456803, −7.30115882129022975819495134651, −7.10421947644500331215620635119, −6.44932738062071186965230047666, −5.25541716917249270647105251496, −4.29706643833271862033738187812, −3.56702534412855749435095641453, −2.79012740984902619266998920516, −1.36597221637941752799478849235,
0.68953420565427865974920611317, 2.19776527116221247972419327822, 3.24853461402140327716126459240, 4.15329954918298847120546764565, 4.95945884258262340209743730382, 5.97292052010385556696729598962, 6.34161385326774547526252487351, 7.60017406881604383759233798517, 8.275498004458781677615796878207, 8.905895298087675900353140398441