| L(s) = 1 | + i·3-s − 4i·7-s − 9-s + 4·11-s + 2i·13-s + 6i·17-s + 4·19-s + 4·21-s − i·27-s + 2·29-s − 4·31-s + 4i·33-s − 2i·37-s − 2·39-s + 2·41-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 1.51i·7-s − 0.333·9-s + 1.20·11-s + 0.554i·13-s + 1.45i·17-s + 0.917·19-s + 0.872·21-s − 0.192i·27-s + 0.371·29-s − 0.718·31-s + 0.696i·33-s − 0.328i·37-s − 0.320·39-s + 0.312·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.095499950\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.095499950\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 + 4iT - 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 + 10iT - 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 - 16T + 71T^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 - 14iT - 97T^{2} \) |
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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
−8.265595995528917876287035575464, −7.66412074848188367680820773233, −6.77086699595135258450263919442, −6.34781079241062403551111301118, −5.31547676740227850415982513295, −4.35083624161163919962925768335, −3.91864028512330890369043211768, −3.31409451590162164628670898524, −1.80690725447072681773555992056, −0.907186975100964522292272236971,
0.75643389249367705915271138896, 1.88897109439793894212810085101, 2.75605359372617745715172798664, 3.44153367426976733450502456089, 4.68658356235272457345085740745, 5.49678042710000296018544707107, 5.93590539549645645863818444537, 6.88369460417687060647464891100, 7.37784477894771144501749447914, 8.330488391983104161778164168589
