| L(s) = 1 | + 2.58i·3-s + 1.51·5-s − i·7-s − 3.69·9-s − 1.15·11-s + (−3.39 − 1.20i)13-s + 3.92i·15-s + 0.0878·17-s − 8.25·19-s + 2.58·21-s + 6.48·23-s − 2.69·25-s − 1.80i·27-s + 2.75i·29-s − 5.46i·31-s + ⋯ |
| L(s) = 1 | + 1.49i·3-s + 0.678·5-s − 0.377i·7-s − 1.23·9-s − 0.346·11-s + (−0.942 − 0.334i)13-s + 1.01i·15-s + 0.0213·17-s − 1.89·19-s + 0.564·21-s + 1.35·23-s − 0.539·25-s − 0.348i·27-s + 0.510i·29-s − 0.981i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.224 + 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.224 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1243681301\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1243681301\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + iT \) |
| 13 | \( 1 + (3.39 + 1.20i)T \) |
| good | 3 | \( 1 - 2.58iT - 3T^{2} \) |
| 5 | \( 1 - 1.51T + 5T^{2} \) |
| 11 | \( 1 + 1.15T + 11T^{2} \) |
| 17 | \( 1 - 0.0878T + 17T^{2} \) |
| 19 | \( 1 + 8.25T + 19T^{2} \) |
| 23 | \( 1 - 6.48T + 23T^{2} \) |
| 29 | \( 1 - 2.75iT - 29T^{2} \) |
| 31 | \( 1 + 5.46iT - 31T^{2} \) |
| 37 | \( 1 + 8.75T + 37T^{2} \) |
| 41 | \( 1 + 0.221iT - 41T^{2} \) |
| 43 | \( 1 + 11.3iT - 43T^{2} \) |
| 47 | \( 1 - 4.58iT - 47T^{2} \) |
| 53 | \( 1 - 10.8iT - 53T^{2} \) |
| 59 | \( 1 + 3.96T + 59T^{2} \) |
| 61 | \( 1 + 11.0iT - 61T^{2} \) |
| 67 | \( 1 + 4.63T + 67T^{2} \) |
| 71 | \( 1 + 9.36iT - 71T^{2} \) |
| 73 | \( 1 + 10.4iT - 73T^{2} \) |
| 79 | \( 1 + 7.68T + 79T^{2} \) |
| 83 | \( 1 + 5.19T + 83T^{2} \) |
| 89 | \( 1 + 1.08iT - 89T^{2} \) |
| 97 | \( 1 - 10.4iT - 97T^{2} \) |
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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
−8.928908386748651209524177432446, −7.890497265360193632207217538338, −7.00547472650923033079133633066, −6.06972571311206388105884405494, −5.23595647855725174034666780225, −4.68466147867435414593255683521, −3.89597769082662193801631041093, −2.96097231701429019487223223022, −1.97261604452941040736812010703, −0.03532260874132731842229973762,
1.49191681070475131381881625217, 2.20454051798277847761109739131, 2.90907644502832466796798887014, 4.42038304615578903847785407734, 5.34649837611314479896467331842, 6.09051604692194511906095798591, 6.84762416691023949549247406933, 7.21690920979625891848187487385, 8.315280065082210702244813408819, 8.665772242529196601698040398729
