| L(s) = 1 | − 6.66i·3-s + 26.7·5-s − 51.4·7-s + 36.5·9-s + 25.0·11-s − 53.2i·13-s − 177. i·15-s − 24.4·17-s + (78.0 − 352. i)19-s + 342. i·21-s + 612.·23-s + 88.0·25-s − 783. i·27-s − 1.34e3i·29-s + 912. i·31-s + ⋯ |
| L(s) = 1 | − 0.740i·3-s + 1.06·5-s − 1.04·7-s + 0.451·9-s + 0.207·11-s − 0.315i·13-s − 0.790i·15-s − 0.0847·17-s + (0.216 − 0.976i)19-s + 0.777i·21-s + 1.15·23-s + 0.140·25-s − 1.07i·27-s − 1.60i·29-s + 0.949i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.216 + 0.976i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.216 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.038620132\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.038620132\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (-78.0 + 352. i)T \) |
| good | 3 | \( 1 + 6.66iT - 81T^{2} \) |
| 5 | \( 1 - 26.7T + 625T^{2} \) |
| 7 | \( 1 + 51.4T + 2.40e3T^{2} \) |
| 11 | \( 1 - 25.0T + 1.46e4T^{2} \) |
| 13 | \( 1 + 53.2iT - 2.85e4T^{2} \) |
| 17 | \( 1 + 24.4T + 8.35e4T^{2} \) |
| 23 | \( 1 - 612.T + 2.79e5T^{2} \) |
| 29 | \( 1 + 1.34e3iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 912. iT - 9.23e5T^{2} \) |
| 37 | \( 1 - 325. iT - 1.87e6T^{2} \) |
| 41 | \( 1 + 51.7iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 2.54e3T + 3.41e6T^{2} \) |
| 47 | \( 1 + 2.99e3T + 4.87e6T^{2} \) |
| 53 | \( 1 + 4.06e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + 1.07e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 6.53e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + 7.38e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + 5.80e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 3.62e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 3.90e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 - 997.T + 4.74e7T^{2} \) |
| 89 | \( 1 + 6.15e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 5.02e3iT - 8.85e7T^{2} \) |
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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
−10.69469728267144639197166842543, −9.727377611324821831144280756540, −9.178701471455976432129640406780, −7.75759366927059296416556756904, −6.68874136652427045221066021431, −6.18091248209188234395495437356, −4.84611067998106154816860466994, −3.17723384506878605079857813077, −1.98658916510893046758531068877, −0.64245885499126082709879578744,
1.43813337732505520109460275133, 2.99111396936433515807883887327, 4.13926614822802868686522213132, 5.40107258472784575332680109355, 6.33121363872112681670391348682, 7.31434539422560278127834882132, 8.974096613401524040054213325610, 9.556555091286255463648393536982, 10.18875914624948982213458095935, 11.06433455041281224319213499362
