L(s) = 1 | − 3i·3-s + 6.24i·5-s + 7·7-s − 9·9-s − 63.4i·11-s + 82.2i·13-s + 18.7·15-s + 75.4·17-s − 125. i·19-s − 21i·21-s − 155.·23-s + 85.9·25-s + 27i·27-s − 56.2i·29-s − 159.·31-s + ⋯ |
L(s) = 1 | − 0.577i·3-s + 0.558i·5-s + 0.377·7-s − 0.333·9-s − 1.73i·11-s + 1.75i·13-s + 0.322·15-s + 1.07·17-s − 1.51i·19-s − 0.218i·21-s − 1.40·23-s + 0.687·25-s + 0.192i·27-s − 0.360i·29-s − 0.922·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.666544366\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.666544366\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + 3iT \) |
| 7 | \( 1 - 7T \) |
good | 5 | \( 1 - 6.24iT - 125T^{2} \) |
| 11 | \( 1 + 63.4iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 82.2iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 75.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 125. iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 155.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 56.2iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 159.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 197. iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 137.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 295. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 186.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 409. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 311. iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 168. iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 563. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 282.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 250.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 948.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.28e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 655.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 706.T + 9.12e5T^{2} \) |
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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
−8.886448970314811891108379592371, −8.202637329371032980008719360022, −7.32767791373316477945630838526, −6.51783777479047833146744663962, −5.91752144365266278466409273661, −4.79981676705828627150007009896, −3.65296076746391529365075977024, −2.73590567119428233334116346455, −1.62377485601993510118963131349, −0.41285418062095750269134669372,
1.11139246096518543949350364191, 2.27094437162044015640937187143, 3.58958590384448169747040149904, 4.35370392222941797132268787179, 5.41056037198173125144187327553, 5.73477185548854364371187721424, 7.34772383715905777276043000272, 7.83116249916194385406163212268, 8.652488861276068065010491702532, 9.657840405660110157897824483678