L(s) = 1 | − 0.935i·2-s + (−0.227 + 1.71i)3-s + 1.12·4-s + (1.60 + 0.212i)6-s − 2.92i·8-s + (−2.89 − 0.781i)9-s + (−0.255 + 1.92i)12-s − 4.88·13-s − 0.488·16-s + (−0.731 + 2.71i)18-s + 4.79i·23-s + (5.02 + 0.665i)24-s − 5·25-s + 4.57i·26-s + (1.99 − 4.79i)27-s + ⋯ |
L(s) = 1 | − 0.661i·2-s + (−0.131 + 0.991i)3-s + 0.561·4-s + (0.656 + 0.0869i)6-s − 1.03i·8-s + (−0.965 − 0.260i)9-s + (−0.0738 + 0.557i)12-s − 1.35·13-s − 0.122·16-s + (−0.172 + 0.638i)18-s + 0.999i·23-s + (1.02 + 0.135i)24-s − 25-s + 0.896i·26-s + (0.384 − 0.922i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.131i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 + 0.131i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.956967 - 0.0631074i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.956967 - 0.0631074i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.227 - 1.71i)T \) |
| 23 | \( 1 - 4.79iT \) |
good | 2 | \( 1 + 0.935iT - 2T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 4.88T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 29 | \( 1 + 8.43iT - 29T^{2} \) |
| 31 | \( 1 - 11.1T + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 - 12.1iT - 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 - 6.55iT - 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 9.59iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 14.0iT - 71T^{2} \) |
| 73 | \( 1 + 7.61T + 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−15.03435518990722677381047890312, −13.61295598795116149295986137262, −12.04724894814315987598064062815, −11.47720881076668553188124078781, −10.15132886490410513069619768099, −9.600630115700918092754364959852, −7.75392894977755580774051116033, −6.07970326932404353267966841481, −4.40022809963226505440061252615, −2.77192614700738942165334686396,
2.42077868263742505761468144597, 5.28722177481878344435313472929, 6.60575718879056758640354074663, 7.44461818042972989246570578362, 8.556597545613138012436785354598, 10.40307041637382338142299505049, 11.75426659891393102706861973863, 12.45215398479418620567206413169, 13.90525843992720231666043891769, 14.70986538228289658088489890541