| L(s) = 1 | − 239. i·5-s − 396.·7-s + 1.96e3i·11-s + 934.·13-s + 1.90e3i·17-s − 9.04e3·19-s + 2.77e3i·23-s − 4.18e4·25-s + 2.46e4i·29-s + 2.38e4·31-s + 9.51e4i·35-s − 3.72e4·37-s − 1.00e5i·41-s − 1.47e5·43-s + 1.02e4i·47-s + ⋯ |
| L(s) = 1 | − 1.91i·5-s − 1.15·7-s + 1.47i·11-s + 0.425·13-s + 0.386i·17-s − 1.31·19-s + 0.228i·23-s − 2.67·25-s + 1.01i·29-s + 0.799·31-s + 2.21i·35-s − 0.736·37-s − 1.45i·41-s − 1.85·43-s + 0.0984i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.0192730 + 0.0606380i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0192730 + 0.0606380i\) |
| \(L(4)\) |
|
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 | \( 1 + 239. iT - 1.56e4T^{2} \) |
| 7 | \( 1 + 396.T + 1.17e5T^{2} \) |
| 11 | \( 1 - 1.96e3iT - 1.77e6T^{2} \) |
| 13 | \( 1 - 934.T + 4.82e6T^{2} \) |
| 17 | \( 1 - 1.90e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 + 9.04e3T + 4.70e7T^{2} \) |
| 23 | \( 1 - 2.77e3iT - 1.48e8T^{2} \) |
| 29 | \( 1 - 2.46e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 - 2.38e4T + 8.87e8T^{2} \) |
| 37 | \( 1 + 3.72e4T + 2.56e9T^{2} \) |
| 41 | \( 1 + 1.00e5iT - 4.75e9T^{2} \) |
| 43 | \( 1 + 1.47e5T + 6.32e9T^{2} \) |
| 47 | \( 1 - 1.02e4iT - 1.07e10T^{2} \) |
| 53 | \( 1 + 1.45e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 - 1.90e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 + 1.23e5T + 5.15e10T^{2} \) |
| 67 | \( 1 - 4.73e5T + 9.04e10T^{2} \) |
| 71 | \( 1 + 1.72e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 9.25e4T + 1.51e11T^{2} \) |
| 79 | \( 1 - 6.25e3T + 2.43e11T^{2} \) |
| 83 | \( 1 + 3.19e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 - 7.40e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 3.48e5T + 8.32e11T^{2} \) |
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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
−12.70023871435822146614793148968, −12.15203346487728732053051458389, −10.21969289207393538895634318006, −9.250022768327023919219299327560, −8.335313313094586898693054361880, −6.67857588228079975407222487890, −5.17832418069192971853823606633, −4.00816903293575151479528925255, −1.69779240618242093923938838719, −0.02352353707224936580317371565,
2.71832180932509790311614856821, 3.59508293480981589893590821480, 6.18987727408803657841336297404, 6.63211429866077335707665630771, 8.232805648326506072720226247893, 9.830091318578660078056101664227, 10.73090148282421517596021572680, 11.58598878616527055962528347203, 13.25297838696025726864851714101, 14.01546237941943632387649081348
