L(s) = 1 | − 10.4·3-s + (−17.9 − 52.9i)5-s − 86.7i·7-s − 134.·9-s − 258. i·11-s + 465.·13-s + (187. + 551. i)15-s − 2.02e3i·17-s − 1.05e3i·19-s + 904. i·21-s − 3.99e3i·23-s + (−2.47e3 + 1.90e3i)25-s + 3.93e3·27-s + 3.24e3i·29-s − 991.·31-s + ⋯ |
L(s) = 1 | − 0.668·3-s + (−0.321 − 0.946i)5-s − 0.669i·7-s − 0.552·9-s − 0.644i·11-s + 0.763·13-s + (0.215 + 0.633i)15-s − 1.70i·17-s − 0.667i·19-s + 0.447i·21-s − 1.57i·23-s + (−0.792 + 0.609i)25-s + 1.03·27-s + 0.715i·29-s − 0.185·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.897 - 0.441i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.897 - 0.441i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.7846100361\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7846100361\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (17.9 + 52.9i)T \) |
good | 3 | \( 1 + 10.4T + 243T^{2} \) |
| 7 | \( 1 + 86.7iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 258. iT - 1.61e5T^{2} \) |
| 13 | \( 1 - 465.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 2.02e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 1.05e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 + 3.99e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 3.24e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 991.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 8.82e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 5.63e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.77e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.77e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 2.59e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.81e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 - 3.54e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 + 4.20e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 7.13e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.25e3iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 8.42e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 7.90e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 6.51e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 3.39e4iT - 8.58e9T^{2} \) |
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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
−10.49967809074661552072135635624, −9.135936077357748798550820176995, −8.515414298180453661258432376705, −7.34005513710988213131386283563, −6.23023294179811914360033238378, −5.19084921205168924953673013439, −4.34645329932659395654533429422, −2.92814594923882754090090341511, −0.923662891989800893663854244573, −0.29025436164926788000856105119,
1.66953869193184611159839925655, 3.07330831542107247368761672870, 4.21185124354377366843858990596, 5.88562973865718196600251173310, 6.09351111345565389624270786137, 7.51782301946266690982836579908, 8.386833301533513510491185837834, 9.611819504955254579009270987559, 10.63725951453031447186108886870, 11.29468876547616043389009691836