| L(s) = 1 | + (0.825 + 1.52i)3-s − 3.94i·5-s + (−1.63 + 2.51i)9-s − 3.52·11-s + 2·13-s + (6.00 − 3.25i)15-s − 5.02i·17-s + 3.04i·19-s − 1.65·23-s − 10.5·25-s + (−5.17 − 0.418i)27-s − 6.80i·29-s + 1.73i·31-s + (−2.91 − 5.37i)33-s + 1.27·37-s + ⋯ |
| L(s) = 1 | + (0.476 + 0.879i)3-s − 1.76i·5-s + (−0.545 + 0.837i)9-s − 1.06·11-s + 0.554·13-s + (1.55 − 0.840i)15-s − 1.21i·17-s + 0.698i·19-s − 0.344·23-s − 2.10·25-s + (−0.996 − 0.0805i)27-s − 1.26i·29-s + 0.311i·31-s + (−0.506 − 0.935i)33-s + 0.209·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.523 + 0.852i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.523 + 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.034760981\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.034760981\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.825 - 1.52i)T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 3.94iT - 5T^{2} \) |
| 11 | \( 1 + 3.52T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 5.02iT - 17T^{2} \) |
| 19 | \( 1 - 3.04iT - 19T^{2} \) |
| 23 | \( 1 + 1.65T + 23T^{2} \) |
| 29 | \( 1 + 6.80iT - 29T^{2} \) |
| 31 | \( 1 - 1.73iT - 31T^{2} \) |
| 37 | \( 1 - 1.27T + 37T^{2} \) |
| 41 | \( 1 + 2.16iT - 41T^{2} \) |
| 43 | \( 1 - 0.837iT - 43T^{2} \) |
| 47 | \( 1 - 4.95T + 47T^{2} \) |
| 53 | \( 1 + 9.66iT - 53T^{2} \) |
| 59 | \( 1 + 10.1T + 59T^{2} \) |
| 61 | \( 1 + 11.2T + 61T^{2} \) |
| 67 | \( 1 + 16.0iT - 67T^{2} \) |
| 71 | \( 1 + 10.3T + 71T^{2} \) |
| 73 | \( 1 + 7.27T + 73T^{2} \) |
| 79 | \( 1 + 5.19iT - 79T^{2} \) |
| 83 | \( 1 + 5.17T + 83T^{2} \) |
| 89 | \( 1 - 7.19iT - 89T^{2} \) |
| 97 | \( 1 - 4.27T + 97T^{2} \) |
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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
−8.862345641671499407910581335229, −8.001407054446807249841596259033, −7.71272519174084658293928486424, −6.04432649308275808607348310743, −5.32552429313470054365433256242, −4.72926702093549718780184990267, −4.04939581381948078377385206097, −2.95627693434053131161675613289, −1.79797490420952672627427517993, −0.30759793058353097412479755842,
1.62005140829730079617523043769, 2.70157624477866184756477434297, 3.12681993498247082063116349885, 4.17684386313725126720453779020, 5.76225349066369237724701547657, 6.20398906997984514828144964878, 7.10287186725647416615715387517, 7.52157404560073970266235108830, 8.308675486077161327831016357896, 9.088643332714115062140874898404
