| L(s) = 1 | + 0.680i·5-s − i·7-s + 1.70·11-s + 5.59·13-s − 3.68i·17-s − 4.46i·19-s + 1.16·23-s + 4.53·25-s + 1.77i·29-s + 2.58i·31-s + 0.680·35-s + 1.36·37-s + 1.31i·41-s − 4.71i·43-s − 3.87·47-s + ⋯ |
| L(s) = 1 | + 0.304i·5-s − 0.377i·7-s + 0.513·11-s + 1.55·13-s − 0.894i·17-s − 1.02i·19-s + 0.242·23-s + 0.907·25-s + 0.329i·29-s + 0.464i·31-s + 0.115·35-s + 0.223·37-s + 0.205i·41-s − 0.718i·43-s − 0.565·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.225452310\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.225452310\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + iT \) |
| good | 5 | \( 1 - 0.680iT - 5T^{2} \) |
| 11 | \( 1 - 1.70T + 11T^{2} \) |
| 13 | \( 1 - 5.59T + 13T^{2} \) |
| 17 | \( 1 + 3.68iT - 17T^{2} \) |
| 19 | \( 1 + 4.46iT - 19T^{2} \) |
| 23 | \( 1 - 1.16T + 23T^{2} \) |
| 29 | \( 1 - 1.77iT - 29T^{2} \) |
| 31 | \( 1 - 2.58iT - 31T^{2} \) |
| 37 | \( 1 - 1.36T + 37T^{2} \) |
| 41 | \( 1 - 1.31iT - 41T^{2} \) |
| 43 | \( 1 + 4.71iT - 43T^{2} \) |
| 47 | \( 1 + 3.87T + 47T^{2} \) |
| 53 | \( 1 + 6.39iT - 53T^{2} \) |
| 59 | \( 1 + 0.948T + 59T^{2} \) |
| 61 | \( 1 + 1.40T + 61T^{2} \) |
| 67 | \( 1 - 1.28iT - 67T^{2} \) |
| 71 | \( 1 + 3.09T + 71T^{2} \) |
| 73 | \( 1 + 5.79T + 73T^{2} \) |
| 79 | \( 1 + 1.72iT - 79T^{2} \) |
| 83 | \( 1 + 7.69T + 83T^{2} \) |
| 89 | \( 1 + 0.962iT - 89T^{2} \) |
| 97 | \( 1 - 12.3T + 97T^{2} \) |
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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
−8.002742494157786552065155456864, −7.02833974420192668186279826618, −6.77270537693122628773492077507, −5.94069770580311479845689480304, −5.05536162465585668859907505990, −4.34099075674743132968982417236, −3.44146378390195373364976503198, −2.85981826188638407340317984516, −1.59338706102986386751699233289, −0.66957987455599207295428966358,
1.07161641328095084941525312788, 1.77138248341515693770124150247, 3.00109167013630908313798154049, 3.79762391284900707086499106029, 4.40697448603256885692742683975, 5.43069287439003193402140736950, 6.13338211073562008813197601535, 6.48187110429071266780368471255, 7.60438217599048707148162212108, 8.259773201547908323272049711792
