| L(s) = 1 | − 2·3-s − 3·5-s + (−2 − 1.73i)7-s + 9-s + 6·11-s + (−1 − 3.46i)13-s + 6·15-s − 3.46i·17-s + 1.73i·19-s + (4 + 3.46i)21-s − 8.66i·23-s + 4·25-s + 4·27-s − 9·29-s − 5.19i·31-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1.34·5-s + (−0.755 − 0.654i)7-s + 0.333·9-s + 1.80·11-s + (−0.277 − 0.960i)13-s + 1.54·15-s − 0.840i·17-s + 0.397i·19-s + (0.872 + 0.755i)21-s − 1.80i·23-s + 0.800·25-s + 0.769·27-s − 1.67·29-s − 0.933i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.576 - 0.817i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.576 - 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (2 + 1.73i)T \) |
| 13 | \( 1 + (1 + 3.46i)T \) |
| good | 3 | \( 1 + 2T + 3T^{2} \) |
| 5 | \( 1 + 3T + 5T^{2} \) |
| 11 | \( 1 - 6T + 11T^{2} \) |
| 17 | \( 1 + 3.46iT - 17T^{2} \) |
| 19 | \( 1 - 1.73iT - 19T^{2} \) |
| 23 | \( 1 + 8.66iT - 23T^{2} \) |
| 29 | \( 1 + 9T + 29T^{2} \) |
| 31 | \( 1 + 5.19iT - 31T^{2} \) |
| 37 | \( 1 - 3.46iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 5.19iT - 43T^{2} \) |
| 47 | \( 1 - 8.66iT - 47T^{2} \) |
| 53 | \( 1 - 3T + 53T^{2} \) |
| 59 | \( 1 - 10.3iT - 59T^{2} \) |
| 61 | \( 1 + 6.92iT - 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 + 7T + 73T^{2} \) |
| 79 | \( 1 + 1.73iT - 79T^{2} \) |
| 83 | \( 1 - 1.73iT - 83T^{2} \) |
| 89 | \( 1 + 3T + 89T^{2} \) |
| 97 | \( 1 - T + 97T^{2} \) |
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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
−9.056860251949111922643930480792, −8.038671384584834801638565192015, −7.21439100933166892135693050450, −6.55779742529963456007433562141, −5.83599478030739669770977041965, −4.60499163799217517037734143761, −3.99344180569757607347819707129, −3.06100076669190010697068461079, −0.901879461596717078397789662509, 0,
1.65481148225513707879807601465, 3.55947952836511305564854544919, 3.93611523571554855113300162461, 5.16320812988207779114694684378, 5.99020455304641038096944892535, 6.80987463965287147273190404107, 7.29184046710677375649099875479, 8.662419596882109551571716342446, 9.117569109669849059824498953562
