| L(s) = 1 | + (0.203 + 1.72i)3-s + 3.44·5-s + i·7-s + (−2.91 + 0.699i)9-s + 5.42i·13-s + (0.699 + 5.91i)15-s − 3.15i·17-s + 4.40·19-s + (−1.72 + 0.203i)21-s − 4.55·23-s + 6.83·25-s + (−1.79 − 4.87i)27-s + 4.08·29-s + 3.44i·35-s + 3.18i·37-s + ⋯ |
| L(s) = 1 | + (0.117 + 0.993i)3-s + 1.53·5-s + 0.377i·7-s + (−0.972 + 0.233i)9-s + 1.50i·13-s + (0.180 + 1.52i)15-s − 0.765i·17-s + 1.01·19-s + (−0.375 + 0.0443i)21-s − 0.949·23-s + 1.36·25-s + (−0.345 − 0.938i)27-s + 0.757·29-s + 0.581i·35-s + 0.523i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.131 - 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.131 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.45322 + 1.27269i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.45322 + 1.27269i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.203 - 1.72i)T \) |
| 7 | \( 1 - iT \) |
| good | 5 | \( 1 - 3.44T + 5T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 5.42iT - 13T^{2} \) |
| 17 | \( 1 + 3.15iT - 17T^{2} \) |
| 19 | \( 1 - 4.40T + 19T^{2} \) |
| 23 | \( 1 + 4.55T + 23T^{2} \) |
| 29 | \( 1 - 4.08T + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 3.18iT - 37T^{2} \) |
| 41 | \( 1 + 5.95iT - 41T^{2} \) |
| 43 | \( 1 + 9.83T + 43T^{2} \) |
| 47 | \( 1 - 10.9T + 47T^{2} \) |
| 53 | \( 1 + 6.31T + 53T^{2} \) |
| 59 | \( 1 - 0.641iT - 59T^{2} \) |
| 61 | \( 1 - 10.2iT - 61T^{2} \) |
| 67 | \( 1 - 9.02T + 67T^{2} \) |
| 71 | \( 1 - 3.15T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 - 1.62iT - 79T^{2} \) |
| 83 | \( 1 + 12.5iT - 83T^{2} \) |
| 89 | \( 1 + 7.80iT - 89T^{2} \) |
| 97 | \( 1 + 16.0T + 97T^{2} \) |
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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
−10.40144746392406629265981814951, −9.730154872164380740535087959136, −9.265270390975864398992991259926, −8.486363473911569412107046858870, −7.00361335007206761603443495222, −6.01038913937918725339230818085, −5.28792618519978176588795868086, −4.35148080762993998079656296637, −2.94566501424293553125045185653, −1.89834086829333791644798490977,
1.10139391566579323654759940400, 2.25565002714985962923630914539, 3.34255373606098110823809476817, 5.19956540179598283120789363122, 5.90814927563528762415863972214, 6.61854925451852386455120299287, 7.74136872148439167486379656263, 8.409430264217890541499231186370, 9.580001852570244728919523918356, 10.19807122570094203207126199376
