L(s) = 1 | + i·3-s − i·5-s − 9-s − 4i·11-s − 2i·13-s + 15-s + 2·17-s − 4i·19-s − 25-s − i·27-s − 2i·29-s + 4·33-s + 10i·37-s + 2·39-s − 10·41-s + ⋯ |
L(s) = 1 | + 0.577i·3-s − 0.447i·5-s − 0.333·9-s − 1.20i·11-s − 0.554i·13-s + 0.258·15-s + 0.485·17-s − 0.917i·19-s − 0.200·25-s − 0.192i·27-s − 0.371i·29-s + 0.696·33-s + 1.64i·37-s + 0.320·39-s − 1.56·41-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
\(L(1)\) |
\(\approx\) |
\(0.7981211110\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7981211110\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 4iT - 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 2iT - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 10iT - 37T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 - 10iT - 53T^{2} \) |
| 59 | \( 1 + 4iT - 59T^{2} \) |
| 61 | \( 1 + 2iT - 61T^{2} \) |
| 67 | \( 1 + 12iT - 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + 10T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 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
−8.321054569842023359020665420379, −7.70937619149048557652213230536, −6.55823237378054144170496292103, −5.94270219899862247246571898096, −5.07863490452185573928430053127, −4.56858212499066697266971008021, −3.36258123886322483064979221256, −2.96841844202890724872308200851, −1.44180050196798747769109932427, −0.22695768056695850267662283784,
1.51346847797634494888823595439, 2.17609484352965320585273225398, 3.29303287193962016839352071227, 4.11320977457367025366339236986, 5.10064812593398518040607489031, 5.85809434502423457454328825604, 6.80408937197220044945956823396, 7.14244581524667954109325995718, 7.939099336993195826249100835102, 8.635568669000205997865447065487