L(s) = 1 | + 5.21i·2-s − 19.2·4-s − 5.83i·5-s − 31.3i·7-s − 58.6i·8-s + 30.4·10-s + 16.2i·11-s + (−43.4 − 17.5i)13-s + 163.·14-s + 152.·16-s − 54·17-s − 66.3i·19-s + 112. i·20-s − 84.9·22-s − 182.·23-s + ⋯ |
L(s) = 1 | + 1.84i·2-s − 2.40·4-s − 0.522i·5-s − 1.69i·7-s − 2.59i·8-s + 0.963·10-s + 0.446i·11-s + (−0.927 − 0.373i)13-s + 3.11·14-s + 2.37·16-s − 0.770·17-s − 0.801i·19-s + 1.25i·20-s − 0.823·22-s − 1.65·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 + 0.373i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.927 + 0.373i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.692905 - 0.134351i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.692905 - 0.134351i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 13 | \( 1 + (43.4 + 17.5i)T \) |
good | 2 | \( 1 - 5.21iT - 8T^{2} \) |
| 5 | \( 1 + 5.83iT - 125T^{2} \) |
| 7 | \( 1 + 31.3iT - 343T^{2} \) |
| 11 | \( 1 - 16.2iT - 1.33e3T^{2} \) |
| 17 | \( 1 + 54T + 4.91e3T^{2} \) |
| 19 | \( 1 + 66.3iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 182.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 164.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 58.9iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 110. iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 55.0iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 113.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 514. iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 242.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 265. iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 468.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 852. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 165. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 315. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 479.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 574. iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 66.7iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 1.43e3iT - 9.12e5T^{2} \) |
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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
−13.49446368952579724867469745596, −12.44659639603886614438660368438, −10.49552246571708860951671917529, −9.496562976691881982089466116712, −8.257761145054209360831077569957, −7.34696308426532355518622749230, −6.58069158454424595187403275761, −4.97310403215082022596600585596, −4.21499655669178767476326044704, −0.37285511647387712587146729768,
2.06578689644116213817655823549, 2.97540791515606460855049990846, 4.59278120986566415526110661767, 6.03209058916779068237053169881, 8.263036900372385501786206717755, 9.222952740416132511091501245603, 10.11290978580980408788101653022, 11.21560084895045645001580920797, 12.05874448827510731504853080398, 12.55592694232754577263899609005