| L(s) = 1 | − 3i·3-s + 4.35·5-s + (−7.09 − 17.1i)7-s − 9·9-s + 43.1·11-s − 26.5·13-s − 13.0i·15-s + 76.6i·17-s − 58.7i·19-s + (−51.3 + 21.2i)21-s − 0.747i·23-s − 106.·25-s + 27i·27-s − 64.9i·29-s − 165.·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 0.389·5-s + (−0.382 − 0.923i)7-s − 0.333·9-s + 1.18·11-s − 0.565·13-s − 0.224i·15-s + 1.09i·17-s − 0.709i·19-s + (−0.533 + 0.221i)21-s − 0.00677i·23-s − 0.848·25-s + 0.192i·27-s − 0.416i·29-s − 0.958·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.924 - 0.382i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.924 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6204693262\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6204693262\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3iT \) |
| 7 | \( 1 + (7.09 + 17.1i)T \) |
| good | 5 | \( 1 - 4.35T + 125T^{2} \) |
| 11 | \( 1 - 43.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 26.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 76.6iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 58.7iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 0.747iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 64.9iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 165.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 389. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 290. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 32.4T + 7.95e4T^{2} \) |
| 47 | \( 1 - 368.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 588. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 471. iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 448.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 707.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 997. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 1.22e3iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 160. iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 1.00e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 1.13e3iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 729. iT - 9.12e5T^{2} \) |
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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
−9.028570639992148108129251236379, −7.70274754073219019835045327896, −7.22002693610198290407406122431, −6.33315307308896155435486959366, −5.70047545203250104581500153412, −4.31663194260821686667287424791, −3.64390577382954093760828066509, −2.28453069572297189699462772545, −1.30712925224877013918746027362, −0.13772510604666323830146097929,
1.55286097163037877396245741936, 2.70294446055440889546722310691, 3.60842929632039784903002234711, 4.69708389504299102771144008667, 5.54035548489580269713149432699, 6.28562681097550683478671039490, 7.14962221245948768787387606835, 8.295210177711129376138101504257, 9.095821295382743222311803736630, 9.660408985868655653856066051097
