| L(s) = 1 | − 45.9i·5-s + 80.5·7-s − 113. i·11-s + 70.2·13-s − 256. i·17-s − 192.·19-s − 543. i·23-s − 1.48e3·25-s + 822. i·29-s − 615.·31-s − 3.70e3i·35-s − 2.35e3·37-s − 502. i·41-s + 164.·43-s − 1.04e3i·47-s + ⋯ |
| L(s) = 1 | − 1.83i·5-s + 1.64·7-s − 0.934i·11-s + 0.415·13-s − 0.885i·17-s − 0.533·19-s − 1.02i·23-s − 2.37·25-s + 0.978i·29-s − 0.640·31-s − 3.02i·35-s − 1.72·37-s − 0.298i·41-s + 0.0890·43-s − 0.471i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.983857592\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.983857592\) |
| \(L(3)\) |
|
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 | \( 1 + 45.9iT - 625T^{2} \) |
| 7 | \( 1 - 80.5T + 2.40e3T^{2} \) |
| 11 | \( 1 + 113. iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 70.2T + 2.85e4T^{2} \) |
| 17 | \( 1 + 256. iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 192.T + 1.30e5T^{2} \) |
| 23 | \( 1 + 543. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 822. iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 615.T + 9.23e5T^{2} \) |
| 37 | \( 1 + 2.35e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + 502. iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 164.T + 3.41e6T^{2} \) |
| 47 | \( 1 + 1.04e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 182. iT - 7.89e6T^{2} \) |
| 59 | \( 1 - 1.81e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 2.85e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 8.89e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + 3.55e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 8.00e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 4.02e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + 1.84e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 4.20e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 1.31e4T + 8.85e7T^{2} \) |
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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
−8.690516685110583856788576846649, −8.219811693524632007762097780392, −7.27764768414556299612562359272, −5.91819434488826991511663183815, −5.07229382339722524152898834666, −4.73542887723469341192216779400, −3.67509104328061516929099178175, −2.03830563908056218072387512439, −1.21179142935074529495674550829, −0.38199149627015960932085926955,
1.65528828227204327106653477516, 2.19753688317464044730301814377, 3.47976849198799988566546485793, 4.28279376011725069587702050335, 5.40093834166365060908851898703, 6.31416298789120660350199947483, 7.17687381534193519417341217446, 7.74514323565159744792354160205, 8.504634960910426321589675529744, 9.722233096814255051488939508134
