| L(s) = 1 | + 1.85i·3-s + 5-s + (−1.29 + 2.30i)7-s − 0.432·9-s − 2.01·11-s + 1.82·13-s + 1.85i·15-s − 1.64i·17-s + 1.26i·19-s + (−4.27 − 2.40i)21-s + 3.19i·23-s + 25-s + 4.75i·27-s + 4.57i·29-s + 0.834·31-s + ⋯ |
| L(s) = 1 | + 1.06i·3-s + 0.447·5-s + (−0.490 + 0.871i)7-s − 0.144·9-s − 0.608·11-s + 0.507·13-s + 0.478i·15-s − 0.399i·17-s + 0.289i·19-s + (−0.931 − 0.525i)21-s + 0.665i·23-s + 0.200·25-s + 0.915i·27-s + 0.849i·29-s + 0.149·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 - 0.248i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.968 - 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.337705453\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.337705453\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + (1.29 - 2.30i)T \) |
| good | 3 | \( 1 - 1.85iT - 3T^{2} \) |
| 11 | \( 1 + 2.01T + 11T^{2} \) |
| 13 | \( 1 - 1.82T + 13T^{2} \) |
| 17 | \( 1 + 1.64iT - 17T^{2} \) |
| 19 | \( 1 - 1.26iT - 19T^{2} \) |
| 23 | \( 1 - 3.19iT - 23T^{2} \) |
| 29 | \( 1 - 4.57iT - 29T^{2} \) |
| 31 | \( 1 - 0.834T + 31T^{2} \) |
| 37 | \( 1 - 7.82iT - 37T^{2} \) |
| 41 | \( 1 + 2.75iT - 41T^{2} \) |
| 43 | \( 1 + 6.74T + 43T^{2} \) |
| 47 | \( 1 + 10.8T + 47T^{2} \) |
| 53 | \( 1 - 8.65iT - 53T^{2} \) |
| 59 | \( 1 + 3.88iT - 59T^{2} \) |
| 61 | \( 1 + 0.285T + 61T^{2} \) |
| 67 | \( 1 - 3.94T + 67T^{2} \) |
| 71 | \( 1 - 11.1iT - 71T^{2} \) |
| 73 | \( 1 + 13.4iT - 73T^{2} \) |
| 79 | \( 1 + 12.8iT - 79T^{2} \) |
| 83 | \( 1 - 7.74iT - 83T^{2} \) |
| 89 | \( 1 + 8.39iT - 89T^{2} \) |
| 97 | \( 1 - 9.32iT - 97T^{2} \) |
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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
−9.484936472050830085969303029284, −8.829783699694908426672161914657, −8.060402304120258500111737610346, −6.94418298984806743270917728441, −6.11952482211067152750970575384, −5.28807370749738545295509486541, −4.77103070175826155827026220870, −3.54624811841145264073303204461, −2.93807301155375087897240457688, −1.64748944028818407301088568840,
0.45169269531767975669578313323, 1.58217397117686320141108689969, 2.56614109686678961497144045373, 3.68199820523541216848186926032, 4.65374204484974805078342566812, 5.76525352382341168465900135194, 6.54878299092799036231334435902, 6.97888226954727733607541269317, 7.905445838689764512102008249285, 8.402334983985153594269141416293
