| L(s) = 1 | − 43.8·5-s − 238i·13-s + 111. i·17-s + 1.29e3·25-s + 1.12e3·29-s − 1.68e3i·37-s + 3.16e3i·41-s − 2.40e3·49-s + 5.31e3·53-s − 2.64e3i·61-s + 1.04e4i·65-s + 1.05e4·73-s − 4.89e3i·85-s − 1.57e4i·89-s − 1.87e4·97-s + ⋯ |
| L(s) = 1 | − 1.75·5-s − 1.40i·13-s + 0.386i·17-s + 2.07·25-s + 1.34·29-s − 1.22i·37-s + 1.88i·41-s − 49-s + 1.89·53-s − 0.709i·61-s + 2.46i·65-s + 1.98·73-s − 0.677i·85-s − 1.98i·89-s − 1.98·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 + 0.169i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.985 + 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.3538691961\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3538691961\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 43.8T + 625T^{2} \) |
| 7 | \( 1 + 2.40e3T^{2} \) |
| 11 | \( 1 + 1.46e4T^{2} \) |
| 13 | \( 1 + 238iT - 2.85e4T^{2} \) |
| 17 | \( 1 - 111. iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 1.30e5T^{2} \) |
| 23 | \( 1 - 2.79e5T^{2} \) |
| 29 | \( 1 - 1.12e3T + 7.07e5T^{2} \) |
| 31 | \( 1 + 9.23e5T^{2} \) |
| 37 | \( 1 + 1.68e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 - 3.16e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 3.41e6T^{2} \) |
| 47 | \( 1 - 4.87e6T^{2} \) |
| 53 | \( 1 - 5.31e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + 1.21e7T^{2} \) |
| 61 | \( 1 + 2.64e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 - 2.01e7T^{2} \) |
| 71 | \( 1 - 2.54e7T^{2} \) |
| 73 | \( 1 - 1.05e4T + 2.83e7T^{2} \) |
| 79 | \( 1 + 3.89e7T^{2} \) |
| 83 | \( 1 + 4.74e7T^{2} \) |
| 89 | \( 1 + 1.57e4iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 1.87e4T + 8.85e7T^{2} \) |
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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
−8.473582296978988089374742020767, −8.117883939327408123880364887083, −7.39200500761098460435510144464, −6.47092876856752099201233419165, −5.31038513283702210577144103362, −4.40117607063412088941936224306, −3.55958149660570249200866395435, −2.76109493493058694139961458588, −0.990332650397682746876906820316, −0.10237640529779271402195796737,
1.04141066028886556959907427846, 2.54318766127922846877473978730, 3.69473416248439531812493915410, 4.30216956220706161239601758286, 5.16012271771329061693568973682, 6.66075198014413905152179670084, 7.07809383770952737324027106417, 8.076769725062912651324143052018, 8.628317497835971983899557935516, 9.518554870954146719319263813270
