L(s) = 1 | − 7.17·3-s + (1.28 − 55.8i)5-s + 146. i·7-s − 191.·9-s + 42.4i·11-s + 605.·13-s + (−9.21 + 401. i)15-s − 409. i·17-s + 2.09e3i·19-s − 1.05e3i·21-s − 3.04e3i·23-s + (−3.12e3 − 143. i)25-s + 3.11e3·27-s − 4.59e3i·29-s + 5.11e3·31-s + ⋯ |
L(s) = 1 | − 0.460·3-s + (0.0229 − 0.999i)5-s + 1.13i·7-s − 0.787·9-s + 0.105i·11-s + 0.993·13-s + (−0.0105 + 0.460i)15-s − 0.343i·17-s + 1.33i·19-s − 0.521i·21-s − 1.20i·23-s + (−0.998 − 0.0459i)25-s + 0.823·27-s − 1.01i·29-s + 0.956·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.961 + 0.273i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.961 + 0.273i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.496404632\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.496404632\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-1.28 + 55.8i)T \) |
good | 3 | \( 1 + 7.17T + 243T^{2} \) |
| 7 | \( 1 - 146. iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 42.4iT - 1.61e5T^{2} \) |
| 13 | \( 1 - 605.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 409. iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 2.09e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 + 3.04e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 4.59e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 5.11e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.12e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.10e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 8.41e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 3.97e3iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 3.82e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.66e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 - 3.18e3iT - 8.44e8T^{2} \) |
| 67 | \( 1 - 4.18e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 7.21e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.82e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 3.94e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 4.65e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 3.82e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 9.59e4iT - 8.58e9T^{2} \) |
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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
−12.01456959363760616786659831503, −11.18593392179090501561816661019, −9.805588422225809974723874648082, −8.712364947766025635943118760376, −8.136017929767346452409330717702, −6.13206802150415068417599365049, −5.61560727497995662042227355326, −4.27350235008974955661003370362, −2.46783801812073663528735592166, −0.78223106472522402994549574091,
0.857438929080818437913635729471, 2.86892139114213569612945741833, 4.08124025999037292172780226244, 5.70514850685648791478419220713, 6.68805837195470221954065644602, 7.64065087038369035869462937528, 8.987376000892122960498813645190, 10.36328839691679418022651117791, 11.01999764782800522103555952978, 11.64628121816894380286486869917