| L(s) = 1 | + 31.6i·5-s + 75.3·7-s − 142. i·11-s − 192.·13-s − 325. i·17-s − 314.·19-s + 512. i·23-s − 377.·25-s + 157. i·29-s + 367.·31-s + 2.38e3i·35-s + 1.73e3·37-s − 395. i·41-s − 720.·43-s + 2.48e3i·47-s + ⋯ |
| L(s) = 1 | + 1.26i·5-s + 1.53·7-s − 1.17i·11-s − 1.13·13-s − 1.12i·17-s − 0.870·19-s + 0.968i·23-s − 0.603·25-s + 0.187i·29-s + 0.382·31-s + 1.94i·35-s + 1.26·37-s − 0.235i·41-s − 0.389·43-s + 1.12i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.127954793\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.127954793\) |
| \(L(3)\) |
|
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 | \( 1 - 31.6iT - 625T^{2} \) |
| 7 | \( 1 - 75.3T + 2.40e3T^{2} \) |
| 11 | \( 1 + 142. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 192.T + 2.85e4T^{2} \) |
| 17 | \( 1 + 325. iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 314.T + 1.30e5T^{2} \) |
| 23 | \( 1 - 512. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 157. iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 367.T + 9.23e5T^{2} \) |
| 37 | \( 1 - 1.73e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + 395. iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 720.T + 3.41e6T^{2} \) |
| 47 | \( 1 - 2.48e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 3.98e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 - 2.46e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 2.48e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 6.59e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 5.82e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 8.79e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 3.86e3T + 3.89e7T^{2} \) |
| 83 | \( 1 - 1.20e4iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 7.63e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 2.91e3T + 8.85e7T^{2} \) |
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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
−9.331628621084881032320001600686, −8.347278244357834343725690584899, −7.62697417838406990073156993804, −7.02227512547247587504073191615, −5.98106298092148999929232451565, −5.11957764772045294267311444571, −4.22147467399716637415422280439, −2.96903970565000807044812528846, −2.32376121742209980545859525485, −0.986309955720871704772111755552,
0.46146057637916057899429194974, 1.68166375462089412067869504324, 2.22601754993809171443677263938, 4.14725601777208794270258831240, 4.71389944869507683601779187251, 5.16999143115910764817207030229, 6.42205916690492670948157699264, 7.49793597392649790490771024233, 8.247962270240052057500740834075, 8.653946859243103917100215897436
