L(s) = 1 | + (0.147 + 0.147i)3-s + 2.61i·5-s + (0.707 + 0.707i)7-s − 2.95i·9-s + (−3.93 − 3.93i)11-s + (−4.00 − 4.00i)13-s + (−0.384 + 0.384i)15-s + (−3.80 + 3.80i)17-s + (2.63 − 2.63i)19-s + 0.207i·21-s − 6.55·23-s − 1.84·25-s + (0.875 − 0.875i)27-s + (−5.02 − 5.02i)29-s + 5.50·31-s + ⋯ |
L(s) = 1 | + (0.0848 + 0.0848i)3-s + 1.16i·5-s + (0.267 + 0.267i)7-s − 0.985i·9-s + (−1.18 − 1.18i)11-s + (−1.11 − 1.11i)13-s + (−0.0993 + 0.0993i)15-s + (−0.923 + 0.923i)17-s + (0.604 − 0.604i)19-s + 0.0453i·21-s − 1.36·23-s − 0.368·25-s + (0.168 − 0.168i)27-s + (−0.933 − 0.933i)29-s + 0.988·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.430 + 0.902i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.430 + 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6299884928\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6299884928\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (-0.707 - 0.707i)T \) |
| 41 | \( 1 + (0.0213 + 6.40i)T \) |
good | 3 | \( 1 + (-0.147 - 0.147i)T + 3iT^{2} \) |
| 5 | \( 1 - 2.61iT - 5T^{2} \) |
| 11 | \( 1 + (3.93 + 3.93i)T + 11iT^{2} \) |
| 13 | \( 1 + (4.00 + 4.00i)T + 13iT^{2} \) |
| 17 | \( 1 + (3.80 - 3.80i)T - 17iT^{2} \) |
| 19 | \( 1 + (-2.63 + 2.63i)T - 19iT^{2} \) |
| 23 | \( 1 + 6.55T + 23T^{2} \) |
| 29 | \( 1 + (5.02 + 5.02i)T + 29iT^{2} \) |
| 31 | \( 1 - 5.50T + 31T^{2} \) |
| 37 | \( 1 - 3.96T + 37T^{2} \) |
| 43 | \( 1 - 5.74iT - 43T^{2} \) |
| 47 | \( 1 + (-3.50 + 3.50i)T - 47iT^{2} \) |
| 53 | \( 1 + (5.19 + 5.19i)T + 53iT^{2} \) |
| 59 | \( 1 - 6.18T + 59T^{2} \) |
| 61 | \( 1 + 1.26iT - 61T^{2} \) |
| 67 | \( 1 + (6.45 - 6.45i)T - 67iT^{2} \) |
| 71 | \( 1 + (-7.03 - 7.03i)T + 71iT^{2} \) |
| 73 | \( 1 - 8.69iT - 73T^{2} \) |
| 79 | \( 1 + (8.32 + 8.32i)T + 79iT^{2} \) |
| 83 | \( 1 + 17.0T + 83T^{2} \) |
| 89 | \( 1 + (6.59 + 6.59i)T + 89iT^{2} \) |
| 97 | \( 1 + (6.38 - 6.38i)T - 97iT^{2} \) |
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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
−9.800368565666904087751406977551, −8.564722521305300886657252584516, −7.937376268581940773426816590013, −7.07208733118708939142214192290, −6.07945781836348405731720821176, −5.51532082485696561786791435320, −4.13867454139318501276669694843, −3.03432612172499560035867563937, −2.46771220789520746865260829560, −0.24991620282011700231987933335,
1.72022630961801231231410117797, 2.51644389984471113853586241900, 4.50489118672386590122645445368, 4.67127478132509410502709343412, 5.54458732151512525639948303962, 7.04786734455120780338230582266, 7.62660646640029583168251449323, 8.308513222227936142444122937402, 9.385144843685523432859201376012, 9.865359765351052217461146179498