L(s) = 1 | + 1.41i·3-s − 5-s − 7-s − 1.00·9-s − 11-s − 1.41i·15-s + 17-s − 19-s − 1.41i·21-s − 1.41i·29-s − 1.41i·31-s − 1.41i·33-s + 35-s + 1.41i·37-s − 1.41i·41-s + ⋯ |
L(s) = 1 | + 1.41i·3-s − 5-s − 7-s − 1.00·9-s − 11-s − 1.41i·15-s + 17-s − 19-s − 1.41i·21-s − 1.41i·29-s − 1.41i·31-s − 1.41i·33-s + 35-s + 1.41i·37-s − 1.41i·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1097172691\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1097172691\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 - 1.41iT - T^{2} \) |
| 5 | \( 1 + T + T^{2} \) |
| 7 | \( 1 + T + T^{2} \) |
| 11 | \( 1 + T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - T + T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + 1.41iT - T^{2} \) |
| 31 | \( 1 + 1.41iT - T^{2} \) |
| 37 | \( 1 - 1.41iT - T^{2} \) |
| 41 | \( 1 + 1.41iT - T^{2} \) |
| 43 | \( 1 - T + T^{2} \) |
| 47 | \( 1 + T + T^{2} \) |
| 53 | \( 1 + 1.41iT - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + T + T^{2} \) |
| 67 | \( 1 + 1.41iT - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( 1 - 1.41iT - T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 - 1.41iT - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.112897033414126146760211438537, −8.115260254411860894282057824453, −7.70415661237614244060168109222, −6.51329234329064185624640400641, −5.68733297317350510191030538498, −4.80315530352820844578770386146, −3.99604739238094352411035670149, −3.48762495864799037793911920137, −2.53042319459580765663924056754, −0.07362187985481995300420422382,
1.39655823689576388873916508275, 2.72447335993537685207080389095, 3.37947586417000630254820142603, 4.53666861509298569748209584891, 5.67964004878423604815519182042, 6.38485427210212267822223132203, 7.23270989218465963473258538260, 7.60186188996864652906585564427, 8.329084354636065712361325121479, 9.081254206913833905609832352449