L(s) = 1 | − 3.14i·5-s + i·7-s − 1.41·11-s + 0.449·13-s + 2.51i·17-s − 2.44i·19-s + 5.65·23-s − 4.89·25-s + 8.34i·29-s − 1.55i·31-s + 3.14·35-s + 9·37-s + 10.0i·41-s − 8.34i·43-s + 0.460·47-s + ⋯ |
L(s) = 1 | − 1.40i·5-s + 0.377i·7-s − 0.426·11-s + 0.124·13-s + 0.608i·17-s − 0.561i·19-s + 1.17·23-s − 0.979·25-s + 1.54i·29-s − 0.278i·31-s + 0.531·35-s + 1.47·37-s + 1.57i·41-s − 1.27i·43-s + 0.0672·47-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
\(L(1)\) |
\(\approx\) |
\(1.917258748\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.917258748\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 3.14iT - 5T^{2} \) |
| 11 | \( 1 + 1.41T + 11T^{2} \) |
| 13 | \( 1 - 0.449T + 13T^{2} \) |
| 17 | \( 1 - 2.51iT - 17T^{2} \) |
| 19 | \( 1 + 2.44iT - 19T^{2} \) |
| 23 | \( 1 - 5.65T + 23T^{2} \) |
| 29 | \( 1 - 8.34iT - 29T^{2} \) |
| 31 | \( 1 + 1.55iT - 31T^{2} \) |
| 37 | \( 1 - 9T + 37T^{2} \) |
| 41 | \( 1 - 10.0iT - 41T^{2} \) |
| 43 | \( 1 + 8.34iT - 43T^{2} \) |
| 47 | \( 1 - 0.460T + 47T^{2} \) |
| 53 | \( 1 - 5.02iT - 53T^{2} \) |
| 59 | \( 1 - 11.4T + 59T^{2} \) |
| 61 | \( 1 + 5.55T + 61T^{2} \) |
| 67 | \( 1 + 13.7iT - 67T^{2} \) |
| 71 | \( 1 + 5.65T + 71T^{2} \) |
| 73 | \( 1 - 9.79T + 73T^{2} \) |
| 79 | \( 1 + 0.348iT - 79T^{2} \) |
| 83 | \( 1 - 2.36T + 83T^{2} \) |
| 89 | \( 1 + 1.55iT - 89T^{2} \) |
| 97 | \( 1 - 10.2T + 97T^{2} \) |
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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
−8.104819931208426985599871833522, −7.40640295827348832231317237007, −6.49369588252490970355083506835, −5.72323674648306090983604667822, −4.98093502809973294038069890211, −4.62810736157576331675689202770, −3.56562142016074721951258866031, −2.64110285411410659860404076209, −1.56488273980748129107658878565, −0.69583990142221778454823065342,
0.793123381242227450985761166101, 2.23799074721963651375769684095, 2.85217292964628235219593144575, 3.63307778956835385829204561790, 4.43780418873700357068986016185, 5.39931615409438729077859727839, 6.16405004093699228066557396953, 6.80349092842131115804932901311, 7.42545428906150725942398162727, 7.924670609996262883176596056451