L(s) = 1 | + i·7-s + 5·11-s − 3i·13-s − i·17-s − 6·19-s − 6i·23-s − 9·29-s − 4·31-s − 2i·37-s + 4·41-s + 10i·43-s − i·47-s − 49-s − 4i·53-s − 8·59-s + ⋯ |
L(s) = 1 | + 0.377i·7-s + 1.50·11-s − 0.832i·13-s − 0.242i·17-s − 1.37·19-s − 1.25i·23-s − 1.67·29-s − 0.718·31-s − 0.328i·37-s + 0.624·41-s + 1.52i·43-s − 0.145i·47-s − 0.142·49-s − 0.549i·53-s − 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5679925395\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5679925395\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - iT \) |
good | 11 | \( 1 - 5T + 11T^{2} \) |
| 13 | \( 1 + 3iT - 13T^{2} \) |
| 17 | \( 1 + iT - 17T^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 + 9T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 - 4T + 41T^{2} \) |
| 43 | \( 1 - 10iT - 43T^{2} \) |
| 47 | \( 1 + iT - 47T^{2} \) |
| 53 | \( 1 + 4iT - 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 + 8T + 61T^{2} \) |
| 67 | \( 1 + 12iT - 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 - 2iT - 73T^{2} \) |
| 79 | \( 1 + 13T + 79T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 - 4T + 89T^{2} \) |
| 97 | \( 1 - 13iT - 97T^{2} \) |
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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
−7.79118981317633536049255396602, −6.96545245413929252753271003510, −6.24849622248479925469332702328, −5.80381895235453923261200190360, −4.75178624535792741349731805971, −4.10861208738486395686089055173, −3.30950576150913836515926375944, −2.33848180506657473760362020051, −1.44663021611817929530769155332, −0.13550983967192206174570100412,
1.42888724028672763865249997475, 1.97846649761999451410251225036, 3.35460838506374001683745741926, 4.04066243561118890196872887829, 4.46687571586783342039539752513, 5.71356802344362859528979674599, 6.15795097553311578530832383953, 7.11972690894318710197543067944, 7.34283278627811604596788535434, 8.478417479436180116409165275107