| L(s) = 1 | + 2.80i·5-s + 7-s + 3.51i·11-s − 4.87i·13-s + 3.16·17-s − 7.87i·19-s + 0.356·23-s − 2.87·25-s + 2.44i·29-s + 7·31-s + 2.80i·35-s + 5.87i·37-s + 10.1·41-s − 8.87i·43-s − 7.34·47-s + ⋯ |
| L(s) = 1 | + 1.25i·5-s + 0.377·7-s + 1.06i·11-s − 1.35i·13-s + 0.766·17-s − 1.80i·19-s + 0.0743·23-s − 0.574·25-s + 0.454i·29-s + 1.25·31-s + 0.474i·35-s + 0.965i·37-s + 1.58·41-s − 1.35i·43-s − 1.07·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.155674874\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.155674874\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| good | 5 | \( 1 - 2.80iT - 5T^{2} \) |
| 11 | \( 1 - 3.51iT - 11T^{2} \) |
| 13 | \( 1 + 4.87iT - 13T^{2} \) |
| 17 | \( 1 - 3.16T + 17T^{2} \) |
| 19 | \( 1 + 7.87iT - 19T^{2} \) |
| 23 | \( 1 - 0.356T + 23T^{2} \) |
| 29 | \( 1 - 2.44iT - 29T^{2} \) |
| 31 | \( 1 - 7T + 31T^{2} \) |
| 37 | \( 1 - 5.87iT - 37T^{2} \) |
| 41 | \( 1 - 10.1T + 41T^{2} \) |
| 43 | \( 1 + 8.87iT - 43T^{2} \) |
| 47 | \( 1 + 7.34T + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 12.9iT - 59T^{2} \) |
| 61 | \( 1 - 1.74iT - 61T^{2} \) |
| 67 | \( 1 + 14.6iT - 67T^{2} \) |
| 71 | \( 1 + 3.51T + 71T^{2} \) |
| 73 | \( 1 + 6.87T + 73T^{2} \) |
| 79 | \( 1 - 10.6T + 79T^{2} \) |
| 83 | \( 1 + 12.9iT - 83T^{2} \) |
| 89 | \( 1 - 10.1T + 89T^{2} \) |
| 97 | \( 1 - 5.74T + 97T^{2} \) |
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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
−7.85210346237590520220180772700, −7.41123957763301505782901332073, −6.74511099829727557114696387142, −6.10831633708939920613581670240, −5.07234229020346439490259192839, −4.64354753907491748302271178878, −3.37017194161124195379231474253, −2.89272636277178192599650976837, −2.05904705587709057306272508772, −0.69525190954627173304912226029,
0.942717416648375686788223624202, 1.54765269465188050733320215034, 2.72682214030002471775119679719, 3.90872543354598798261707271047, 4.31640117089909749497990616992, 5.23521424135846750295982520867, 5.87398484461037390998586654615, 6.44866636492399260327789056208, 7.66578610254732443470083138949, 8.060046634798059518693042764264
