L(s) = 1 | − 3.21i·5-s + 1.36i·11-s − 2.93i·13-s − 6.91i·17-s + 7.35·19-s + 3.62i·23-s − 5.33·25-s + 1.11·29-s + 8.70·31-s + 7.63·37-s + 0.833i·41-s + 4.82i·43-s − 2.95·47-s + 4.57·53-s + 4.39·55-s + ⋯ |
L(s) = 1 | − 1.43i·5-s + 0.412i·11-s − 0.812i·13-s − 1.67i·17-s + 1.68·19-s + 0.756i·23-s − 1.06·25-s + 0.206·29-s + 1.56·31-s + 1.25·37-s + 0.130i·41-s + 0.735i·43-s − 0.430·47-s + 0.628·53-s + 0.592·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.101 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.101 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.210219217\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.210219217\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
good | 5 | \( 1 + 3.21iT - 5T^{2} \) |
| 11 | \( 1 - 1.36iT - 11T^{2} \) |
| 13 | \( 1 + 2.93iT - 13T^{2} \) |
| 17 | \( 1 + 6.91iT - 17T^{2} \) |
| 19 | \( 1 - 7.35T + 19T^{2} \) |
| 23 | \( 1 - 3.62iT - 23T^{2} \) |
| 29 | \( 1 - 1.11T + 29T^{2} \) |
| 31 | \( 1 - 8.70T + 31T^{2} \) |
| 37 | \( 1 - 7.63T + 37T^{2} \) |
| 41 | \( 1 - 0.833iT - 41T^{2} \) |
| 43 | \( 1 - 4.82iT - 43T^{2} \) |
| 47 | \( 1 + 2.95T + 47T^{2} \) |
| 53 | \( 1 - 4.57T + 53T^{2} \) |
| 59 | \( 1 - 14.0T + 59T^{2} \) |
| 61 | \( 1 - 11.0iT - 61T^{2} \) |
| 67 | \( 1 + 12.1iT - 67T^{2} \) |
| 71 | \( 1 + 12.6iT - 71T^{2} \) |
| 73 | \( 1 + 6.49iT - 73T^{2} \) |
| 79 | \( 1 - 7.79iT - 79T^{2} \) |
| 83 | \( 1 + 8.87T + 83T^{2} \) |
| 89 | \( 1 + 12.6iT - 89T^{2} \) |
| 97 | \( 1 - 13.8iT - 97T^{2} \) |
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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
−7.72164977954857806990913913545, −7.31968271241587675780290838579, −6.25225638606545702655521338343, −5.37394928053242055351301119568, −5.00741635101150358621443652759, −4.39075223549704732732998388464, −3.28299654484618887006305156828, −2.55641745632363535304130445731, −1.20034081092970956479154574615, −0.68872666997758125210417531654,
1.06047251964743364901601701237, 2.25012989504867609492030004418, 2.93099879418250677302641874321, 3.71519467565717389394924328198, 4.37093803211219393715643167666, 5.51070640334171377144443880045, 6.14265814080574739412678389456, 6.77245140310074892147786727547, 7.23793425581885885349382166591, 8.193225700408318562366592177658