| L(s) = 1 | + i·3-s − i·5-s − 9-s + 2i·13-s + 15-s + 6·17-s + 4i·19-s + 8·23-s − 25-s − i·27-s − 2i·29-s − 4·31-s − 10i·37-s − 2·39-s − 2·41-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 0.447i·5-s − 0.333·9-s + 0.554i·13-s + 0.258·15-s + 1.45·17-s + 0.917i·19-s + 1.66·23-s − 0.200·25-s − 0.192i·27-s − 0.371i·29-s − 0.718·31-s − 1.64i·37-s − 0.320·39-s − 0.312·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.911007854\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.911007854\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + iT \) |
| good | 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 - 8T + 23T^{2} \) |
| 29 | \( 1 + 2iT - 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + 10iT - 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 - 2iT - 53T^{2} \) |
| 59 | \( 1 - 8iT - 59T^{2} \) |
| 61 | \( 1 + 2iT - 61T^{2} \) |
| 67 | \( 1 - 12iT - 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 - 14T + 73T^{2} \) |
| 79 | \( 1 - 12T + 79T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 - 14T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−8.669941201542239558134397682084, −7.897620318788888414399455137568, −7.21676501343697961242788980754, −6.24561977399359378496670567404, −5.41279020010056978274654825318, −4.93857315295163951926553406596, −3.86888290256714392128786250981, −3.34628195851288466550937329442, −2.09303570358724683866335201499, −0.941266393337892394198209854188,
0.72136183299826268589516836633, 1.78581267250210453233796024109, 3.11109220898003203078983713960, 3.28794000473019828584181593356, 4.89302064282077834299802456525, 5.29591713244453053295412388127, 6.44905454539681158935985290709, 6.80793980648533202355979558551, 7.76196715124696625764126378178, 8.130291947411410281772056904157
