L(s) = 1 | + (−0.5 + 2.59i)7-s − 5.19i·13-s − 7·19-s + 5·25-s + 4·31-s − 11·37-s − 10.3i·43-s + (−6.5 − 2.59i)49-s − 15.5i·61-s + 15.5i·67-s − 15.5i·73-s − 5.19i·79-s + (13.5 + 2.59i)91-s − 5.19i·97-s + 13·103-s + ⋯ |
L(s) = 1 | + (−0.188 + 0.981i)7-s − 1.44i·13-s − 1.60·19-s + 25-s + 0.718·31-s − 1.80·37-s − 1.58i·43-s + (−0.928 − 0.371i)49-s − 1.99i·61-s + 1.90i·67-s − 1.82i·73-s − 0.584i·79-s + (1.41 + 0.272i)91-s − 0.527i·97-s + 1.28·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.188 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.188 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9405976165\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9405976165\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (0.5 - 2.59i)T \) |
good | 5 | \( 1 - 5T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + 5.19iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 7T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 11T + 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 10.3iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 15.5iT - 61T^{2} \) |
| 67 | \( 1 - 15.5iT - 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + 15.5iT - 73T^{2} \) |
| 79 | \( 1 + 5.19iT - 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 + 5.19iT - 97T^{2} \) |
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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
−8.623476487465571274226712558019, −7.926677556188623053776491347935, −6.90577504287534565230776954988, −6.22026805230902556378040357776, −5.43459385225496633594064220032, −4.79514110234140824617061367210, −3.60095581417708514291194309067, −2.80670669404334344327325386113, −1.89662193330354235609065944471, −0.30031630577067337599482413903,
1.22956292702983013668428263382, 2.30259967256612242491342171503, 3.47203153013234921401297197867, 4.32589373637895868643257683056, 4.80798341981885920811238398123, 6.13898765654880443285069397969, 6.73050437750008015806638949079, 7.24510965040544038635997531694, 8.302491159645706900020484982475, 8.853081095465138699184656903214