| L(s) = 1 | − 0.0947·3-s − 5-s + 0.826·7-s − 2.99·9-s − 2.99·13-s + 0.0947·15-s − 0.662·19-s − 0.0783·21-s − 0.473·23-s + 25-s + 0.567·27-s + 4·29-s + 4.64·31-s − 0.826·35-s + 5.98·37-s + 0.283·39-s − 0.394·41-s + 6.96·43-s + 2.99·45-s + 3.88·47-s − 6.31·49-s − 0.801·53-s + 0.0628·57-s + 1.71·59-s + 2.58·61-s − 2.47·63-s + 2.99·65-s + ⋯ |
| L(s) = 1 | − 0.0547·3-s − 0.447·5-s + 0.312·7-s − 0.997·9-s − 0.829·13-s + 0.0244·15-s − 0.152·19-s − 0.0171·21-s − 0.0986·23-s + 0.200·25-s + 0.109·27-s + 0.742·29-s + 0.834·31-s − 0.139·35-s + 0.983·37-s + 0.0453·39-s − 0.0616·41-s + 1.06·43-s + 0.445·45-s + 0.567·47-s − 0.902·49-s − 0.110·53-s + 0.00831·57-s + 0.223·59-s + 0.330·61-s − 0.311·63-s + 0.370·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 0.0947T + 3T^{2} \) |
| 7 | \( 1 - 0.826T + 7T^{2} \) |
| 13 | \( 1 + 2.99T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 0.662T + 19T^{2} \) |
| 23 | \( 1 + 0.473T + 23T^{2} \) |
| 29 | \( 1 - 4T + 29T^{2} \) |
| 31 | \( 1 - 4.64T + 31T^{2} \) |
| 37 | \( 1 - 5.98T + 37T^{2} \) |
| 41 | \( 1 + 0.394T + 41T^{2} \) |
| 43 | \( 1 - 6.96T + 43T^{2} \) |
| 47 | \( 1 - 3.88T + 47T^{2} \) |
| 53 | \( 1 + 0.801T + 53T^{2} \) |
| 59 | \( 1 - 1.71T + 59T^{2} \) |
| 61 | \( 1 - 2.58T + 61T^{2} \) |
| 67 | \( 1 + 5.69T + 67T^{2} \) |
| 71 | \( 1 - 5.59T + 71T^{2} \) |
| 73 | \( 1 - 8T + 73T^{2} \) |
| 79 | \( 1 + 11.6T + 79T^{2} \) |
| 83 | \( 1 + 5.90T + 83T^{2} \) |
| 89 | \( 1 - 4.32T + 89T^{2} \) |
| 97 | \( 1 - 6.50T + 97T^{2} \) |
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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
−7.44175805105501384523969919208, −6.63934058696791089135657305147, −5.99495504606142971205592784392, −5.22266987273651314427025890120, −4.60570777167381269508270313977, −3.88043908233057128780939454243, −2.85913156468548996137064698707, −2.40893930675955047539061388361, −1.08980022643092505653198845159, 0,
1.08980022643092505653198845159, 2.40893930675955047539061388361, 2.85913156468548996137064698707, 3.88043908233057128780939454243, 4.60570777167381269508270313977, 5.22266987273651314427025890120, 5.99495504606142971205592784392, 6.63934058696791089135657305147, 7.44175805105501384523969919208
