| L(s) = 1 | + 19.1i·5-s + 6.40i·7-s + 67.7·11-s + 154. i·13-s − 284.·17-s + 434.·19-s + 462. i·23-s + 259.·25-s + 350. i·29-s + 88.4i·31-s − 122.·35-s − 1.63e3i·37-s − 395.·41-s − 350.·43-s + 2.45e3i·47-s + ⋯ |
| L(s) = 1 | + 0.764i·5-s + 0.130i·7-s + 0.560·11-s + 0.913i·13-s − 0.982·17-s + 1.20·19-s + 0.874i·23-s + 0.415·25-s + 0.416i·29-s + 0.0920i·31-s − 0.0998·35-s − 1.19i·37-s − 0.235·41-s − 0.189·43-s + 1.11i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.706061735\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.706061735\) |
| \(L(3)\) |
|
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 | \( 1 - 19.1iT - 625T^{2} \) |
| 7 | \( 1 - 6.40iT - 2.40e3T^{2} \) |
| 11 | \( 1 - 67.7T + 1.46e4T^{2} \) |
| 13 | \( 1 - 154. iT - 2.85e4T^{2} \) |
| 17 | \( 1 + 284.T + 8.35e4T^{2} \) |
| 19 | \( 1 - 434.T + 1.30e5T^{2} \) |
| 23 | \( 1 - 462. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 350. iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 88.4iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 1.63e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 + 395.T + 2.82e6T^{2} \) |
| 43 | \( 1 + 350.T + 3.41e6T^{2} \) |
| 47 | \( 1 - 2.45e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 3.82e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 - 1.75e3T + 1.21e7T^{2} \) |
| 61 | \( 1 + 2.33e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 + 2.19e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + 4.46e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 2.59e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 2.04e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 - 5.24e3T + 4.74e7T^{2} \) |
| 89 | \( 1 + 1.43e4T + 6.27e7T^{2} \) |
| 97 | \( 1 - 1.68e4T + 8.85e7T^{2} \) |
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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
−9.290925502513702584382181433893, −9.066390151743921166375620676632, −7.73237194607050356189703098787, −7.01931306215369574782808703838, −6.36995808307052061258838222364, −5.35252004866226971529244712571, −4.27882258773985932215875146564, −3.36628355709010804311225004565, −2.34267038123296095607247541491, −1.21970200866152641141479838817,
0.38054921977764347048888840878, 1.26031565627607207448514392972, 2.58003798698651334796783794968, 3.69958263981695866852135903655, 4.69723025679790027142488664086, 5.40251952143205292176882769227, 6.45515472654424069674444253674, 7.26789994683321958807365900828, 8.332283840078047951868740063131, 8.790822928233064087939118076661
