| L(s) = 1 | + 3-s + (−2 + i)5-s + i·7-s + 9-s − 4i·11-s + 2·13-s + (−2 + i)15-s + 2i·17-s + 8i·19-s + i·21-s − 4i·23-s + (3 − 4i)25-s + 27-s + 2i·29-s + 4·31-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + (−0.894 + 0.447i)5-s + 0.377i·7-s + 0.333·9-s − 1.20i·11-s + 0.554·13-s + (−0.516 + 0.258i)15-s + 0.485i·17-s + 1.83i·19-s + 0.218i·21-s − 0.834i·23-s + (0.600 − 0.800i)25-s + 0.192·27-s + 0.371i·29-s + 0.718·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.316 - 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.316 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.737996728\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.737996728\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + (2 - i)T \) |
| 7 | \( 1 - iT \) |
| good | 11 | \( 1 + 4iT - 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 - 8iT - 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 - 2iT - 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 10T + 43T^{2} \) |
| 47 | \( 1 - 6iT - 47T^{2} \) |
| 53 | \( 1 + 14T + 53T^{2} \) |
| 59 | \( 1 - 6iT - 59T^{2} \) |
| 61 | \( 1 - 2iT - 61T^{2} \) |
| 67 | \( 1 + 10T + 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 - 6iT - 73T^{2} \) |
| 79 | \( 1 - 14T + 79T^{2} \) |
| 83 | \( 1 - 12T + 83T^{2} \) |
| 89 | \( 1 - 8T + 89T^{2} \) |
| 97 | \( 1 - 6iT - 97T^{2} \) |
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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
−8.567819942799249714602010885128, −8.098669702489921424113953329183, −7.55835377641132013487815061062, −6.34892621300295132023060141861, −6.05003892972168458098315915763, −4.81489010576000963676604277736, −3.76595964921642079534572064373, −3.41455003614794314970890911207, −2.41801035713232971077379923375, −1.10170041188640126478914951421,
0.56381971390923017164732230783, 1.81906811780195090324792399110, 2.95233662161984075967218719510, 3.78978205021579893172733102290, 4.61824952685902356533292830462, 5.06674516919571137340226387986, 6.48689932135419456135463669843, 7.19264089521570009848293430474, 7.68130236284686718342587077275, 8.426336635636305127068186430565
