L(s) = 1 | + 0.896i·5-s − i·7-s + 3.34·11-s + 4·13-s + 3.48i·17-s + i·19-s + 0.138·23-s + 4.19·25-s + 2.07i·29-s + 0.267i·31-s + 0.896·35-s − 8.26·37-s + 9.28i·41-s + 0.535i·43-s + 5.93·47-s + ⋯ |
L(s) = 1 | + 0.400i·5-s − 0.377i·7-s + 1.00·11-s + 1.10·13-s + 0.845i·17-s + 0.229i·19-s + 0.0289·23-s + 0.839·25-s + 0.384i·29-s + 0.0481i·31-s + 0.151·35-s − 1.35·37-s + 1.44i·41-s + 0.0817i·43-s + 0.865·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.255592247\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.255592247\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + iT \) |
good | 5 | \( 1 - 0.896iT - 5T^{2} \) |
| 11 | \( 1 - 3.34T + 11T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 17 | \( 1 - 3.48iT - 17T^{2} \) |
| 19 | \( 1 - iT - 19T^{2} \) |
| 23 | \( 1 - 0.138T + 23T^{2} \) |
| 29 | \( 1 - 2.07iT - 29T^{2} \) |
| 31 | \( 1 - 0.267iT - 31T^{2} \) |
| 37 | \( 1 + 8.26T + 37T^{2} \) |
| 41 | \( 1 - 9.28iT - 41T^{2} \) |
| 43 | \( 1 - 0.535iT - 43T^{2} \) |
| 47 | \( 1 - 5.93T + 47T^{2} \) |
| 53 | \( 1 + 2.82iT - 53T^{2} \) |
| 59 | \( 1 + 1.03T + 59T^{2} \) |
| 61 | \( 1 - 8.39T + 61T^{2} \) |
| 67 | \( 1 + 2.92iT - 67T^{2} \) |
| 71 | \( 1 - 1.55T + 71T^{2} \) |
| 73 | \( 1 - 6.39T + 73T^{2} \) |
| 79 | \( 1 + 0.535iT - 79T^{2} \) |
| 83 | \( 1 + 12.6T + 83T^{2} \) |
| 89 | \( 1 + 10.7iT - 89T^{2} \) |
| 97 | \( 1 + 10.9T + 97T^{2} \) |
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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
−8.341614423302238742825226659303, −7.33408627950546573042265696578, −6.66951953799077369525195877714, −6.22626451062727744807128599723, −5.36994280828681837047760416270, −4.36903787924627294122337912507, −3.71130790043618205693535224500, −3.09763536587661317739924617664, −1.80167273418313417034843478182, −1.03671576512556810368373951686,
0.69069654014593368674084607377, 1.61883407155371568127029743921, 2.67713297597380962947425978184, 3.63428451419099202405294900562, 4.26275848952264622575817298516, 5.22026063140596292140902650626, 5.75607935970639950548268537722, 6.69776360129655908820600740525, 7.08867130346602775947286838708, 8.155571843365050006472495970323