L(s) = 1 | + (−5.65 − 5.65i)3-s + (−54.5 − 12.2i)5-s + (23.8 − 23.8i)7-s − 178. i·9-s − 218. i·11-s + (152. − 152. i)13-s + (239. + 377. i)15-s + (318. + 318. i)17-s − 2.45e3·19-s − 270.·21-s + (−512. − 512. i)23-s + (2.82e3 + 1.33e3i)25-s + (−2.38e3 + 2.38e3i)27-s + 5.94e3i·29-s + 3.06e3i·31-s + ⋯ |
L(s) = 1 | + (−0.362 − 0.362i)3-s + (−0.975 − 0.218i)5-s + (0.184 − 0.184i)7-s − 0.736i·9-s − 0.543i·11-s + (0.250 − 0.250i)13-s + (0.274 + 0.433i)15-s + (0.267 + 0.267i)17-s − 1.56·19-s − 0.133·21-s + (−0.201 − 0.201i)23-s + (0.904 + 0.426i)25-s + (−0.630 + 0.630i)27-s + 1.31i·29-s + 0.572i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.327 - 0.944i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.327 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.2031982984\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2031982984\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (54.5 + 12.2i)T \) |
good | 3 | \( 1 + (5.65 + 5.65i)T + 243iT^{2} \) |
| 7 | \( 1 + (-23.8 + 23.8i)T - 1.68e4iT^{2} \) |
| 11 | \( 1 + 218. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + (-152. + 152. i)T - 3.71e5iT^{2} \) |
| 17 | \( 1 + (-318. - 318. i)T + 1.41e6iT^{2} \) |
| 19 | \( 1 + 2.45e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + (512. + 512. i)T + 6.43e6iT^{2} \) |
| 29 | \( 1 - 5.94e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 3.06e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 + (-1.31e3 - 1.31e3i)T + 6.93e7iT^{2} \) |
| 41 | \( 1 - 6.09e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + (1.90e3 + 1.90e3i)T + 1.47e8iT^{2} \) |
| 47 | \( 1 + (8.04e3 - 8.04e3i)T - 2.29e8iT^{2} \) |
| 53 | \( 1 + (8.09e3 - 8.09e3i)T - 4.18e8iT^{2} \) |
| 59 | \( 1 + 4.16e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.30e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + (4.16e4 - 4.16e4i)T - 1.35e9iT^{2} \) |
| 71 | \( 1 - 2.37e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + (-9.33e3 + 9.33e3i)T - 2.07e9iT^{2} \) |
| 79 | \( 1 - 8.60e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + (7.58e4 + 7.58e4i)T + 3.93e9iT^{2} \) |
| 89 | \( 1 - 1.86e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 + (9.77e4 + 9.77e4i)T + 8.58e9iT^{2} \) |
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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
−12.38083367578473348642282786318, −11.31593902773784012962868887646, −10.58309178915557732961039435681, −8.985791723321332173726275077215, −8.162649870892493268103780005965, −6.99514316700601065257057660552, −5.95887373528243185852748298491, −4.45272523790869236097599612995, −3.30197122661423958202731523617, −1.17902911520192284746270255018,
0.07853856617321036398884190249, 2.21195161939312623989040451545, 3.96306213843512112418577752820, 4.84984879584204614728079019114, 6.29314190778181634666137027475, 7.59467141091238567406657100949, 8.405533603099569304854176718766, 9.810557974241622678315088807493, 10.85122196745045760150584306232, 11.52588917452838317190659810524