L(s) = 1 | + 8.29i·2-s + 20.3i·3-s − 36.8·4-s + (55.0 + 9.59i)5-s − 168.·6-s + 101. i·7-s − 39.8i·8-s − 169.·9-s + (−79.5 + 456. i)10-s − 362.·11-s − 747. i·12-s − 540. i·13-s − 845.·14-s + (−194. + 1.11e3i)15-s − 847.·16-s + 561. i·17-s + ⋯ |
L(s) = 1 | + 1.46i·2-s + 1.30i·3-s − 1.15·4-s + (0.985 + 0.171i)5-s − 1.91·6-s + 0.786i·7-s − 0.219i·8-s − 0.698·9-s + (−0.251 + 1.44i)10-s − 0.903·11-s − 1.49i·12-s − 0.886i·13-s − 1.15·14-s + (−0.223 + 1.28i)15-s − 0.827·16-s + 0.471i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.171 + 0.985i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.171 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.724851075\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.724851075\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-55.0 - 9.59i)T \) |
| 23 | \( 1 + 529iT \) |
good | 2 | \( 1 - 8.29iT - 32T^{2} \) |
| 3 | \( 1 - 20.3iT - 243T^{2} \) |
| 7 | \( 1 - 101. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 362.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 540. iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 561. iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 1.11e3T + 2.47e6T^{2} \) |
| 29 | \( 1 - 7.26e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 450.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 4.96e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 7.55e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.60e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 1.94e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 5.35e3iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 4.93e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.63e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.44e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 5.39e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.24e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 4.43e3T + 3.07e9T^{2} \) |
| 83 | \( 1 - 9.18e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 8.87e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 5.56e4iT - 8.58e9T^{2} \) |
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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
−13.78967302429692143960492154024, −12.67172748455404224041258252228, −10.81879547056388132248357173632, −10.04942344640460667265339055995, −8.964950815110959858449691136681, −8.074439358485842620303104597548, −6.42108048482741060673798472673, −5.52656829008007783805803883415, −4.72517758308867770575037365780, −2.70189249286172917619152542473,
0.62931305016555509234678932482, 1.71914438263393115023555760907, 2.66246861983355665848731167472, 4.53480990403258999517752986462, 6.32691510820317620921661145660, 7.38218327996330290651196602103, 8.896503689572419184503153731089, 10.08197249101645712285869536207, 10.85161134779635722526557843428, 12.09142181119958421857956983013