L(s) = 1 | + (−5.30 + 1.95i)2-s − 7.37·3-s + (24.3 − 20.7i)4-s + 74.1·5-s + (39.1 − 14.4i)6-s − 80.5i·7-s + (−88.5 + 157. i)8-s − 188.·9-s + (−393. + 145. i)10-s + 299. i·11-s + (−179. + 153. i)12-s − 472. i·13-s + (157. + 427. i)14-s − 546.·15-s + (161. − 1.01e3i)16-s + 861.·17-s + ⋯ |
L(s) = 1 | + (−0.938 + 0.345i)2-s − 0.473·3-s + (0.760 − 0.648i)4-s + 1.32·5-s + (0.443 − 0.163i)6-s − 0.621i·7-s + (−0.489 + 0.872i)8-s − 0.776·9-s + (−1.24 + 0.458i)10-s + 0.747i·11-s + (−0.359 + 0.306i)12-s − 0.775i·13-s + (0.214 + 0.582i)14-s − 0.627·15-s + (0.157 − 0.987i)16-s + 0.722·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.591 + 0.806i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.591 + 0.806i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.911993 - 0.462067i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.911993 - 0.462067i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (5.30 - 1.95i)T \) |
| 19 | \( 1 + (-115. + 1.56e3i)T \) |
good | 3 | \( 1 + 7.37T + 243T^{2} \) |
| 5 | \( 1 - 74.1T + 3.12e3T^{2} \) |
| 7 | \( 1 + 80.5iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 299. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + 472. iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 861.T + 1.41e6T^{2} \) |
| 23 | \( 1 + 840. iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 5.15e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 3.42e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 4.13e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 1.96e4iT - 1.15e8T^{2} \) |
| 43 | \( 1 + 1.19e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 1.58e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 4.88e3iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 3.14e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.94e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.17e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.65e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.62e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.59e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.50e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 6.13e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 - 9.09e4iT - 8.58e9T^{2} \) |
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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
−13.55308124950234351034985839822, −12.06294027109846311870797948860, −10.70491274817853382181378427940, −10.04262500178904910658567058817, −8.948274551695323053500595736135, −7.48619694932313260944004594508, −6.22308217527472471906546035343, −5.28441553318858006821417560533, −2.40469813338601967775998288441, −0.66726982752938189616975359748,
1.43341559530557633738355811361, 2.93390890785525698275771773800, 5.60593995611278185147822527976, 6.43203749975245075656429417111, 8.274212392825912752477138277576, 9.280320314074944958974334646632, 10.22853625304751658130985324184, 11.38851251092187056450358215648, 12.24469532217716347862541720433, 13.61424161545622082719741864352