L(s) = 1 | − 3i·3-s − 20.9i·5-s − 7·7-s − 9·9-s + 10.4i·11-s − 50.9i·13-s − 62.9·15-s + 19.1·17-s + 24.3i·19-s + 21i·21-s + 101.·23-s − 315.·25-s + 27i·27-s − 179. i·29-s − 264.·31-s + ⋯ |
L(s) = 1 | − 0.577i·3-s − 1.87i·5-s − 0.377·7-s − 0.333·9-s + 0.285i·11-s − 1.08i·13-s − 1.08·15-s + 0.273·17-s + 0.294i·19-s + 0.218i·21-s + 0.921·23-s − 2.52·25-s + 0.192i·27-s − 1.14i·29-s − 1.53·31-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
\(L(2)\) |
\(\approx\) |
\(0.7171663797\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7171663797\) |
\(L(\frac{5}{2})\) |
|
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 | \( 1 + 20.9iT - 125T^{2} \) |
| 11 | \( 1 - 10.4iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 50.9iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 19.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 24.3iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 101.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 179. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 264.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 328. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 89.3T + 6.89e4T^{2} \) |
| 43 | \( 1 + 124. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 446.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 384. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 94.7iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 376. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 338. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 268.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 634.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.13e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 589. iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 459.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.36e3T + 9.12e5T^{2} \) |
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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
−8.683599109351855085523292592968, −7.900753866243533363757443130692, −7.25434151094824702855503680619, −5.88946864983188103325344766313, −5.43856941932418460942539908372, −4.49694391203964367325582128414, −3.43590798059250416138082768583, −2.04000465158195244157031644874, −1.00358688339440592851186036524, −0.18090339128964793406990801928,
1.87007374370613425364718868953, 3.13582074943875581694863948035, 3.42664790675945919235322692474, 4.69006805316304043344534914837, 5.79804779542507730631273301518, 6.78204495581645563730250590421, 7.00061613845671196511564367151, 8.199594538837931700296529170893, 9.251656723097299381840368395647, 9.836501035400043441012800563537