L(s) = 1 | + (−0.130 − 0.991i)2-s + (−0.965 + 0.258i)4-s + (1.95 + 0.128i)5-s + (0.382 + 0.923i)8-s + (−0.793 − 0.608i)9-s + (−0.128 − 1.95i)10-s + (1 + i)13-s + (0.866 − 0.5i)16-s + (0.608 + 0.793i)17-s + (−0.499 + 0.866i)18-s + (−1.92 + 0.382i)20-s + (2.82 + 0.371i)25-s + (0.860 − 1.12i)26-s + (−1.08 + 1.63i)29-s + (−0.608 − 0.793i)32-s + ⋯ |
L(s) = 1 | + (−0.130 − 0.991i)2-s + (−0.965 + 0.258i)4-s + (1.95 + 0.128i)5-s + (0.382 + 0.923i)8-s + (−0.793 − 0.608i)9-s + (−0.128 − 1.95i)10-s + (1 + i)13-s + (0.866 − 0.5i)16-s + (0.608 + 0.793i)17-s + (−0.499 + 0.866i)18-s + (−1.92 + 0.382i)20-s + (2.82 + 0.371i)25-s + (0.860 − 1.12i)26-s + (−1.08 + 1.63i)29-s + (−0.608 − 0.793i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.710 + 0.703i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.710 + 0.703i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.529328607\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.529328607\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.130 + 0.991i)T \) |
| 7 | \( 1 \) |
| 17 | \( 1 + (-0.608 - 0.793i)T \) |
good | 3 | \( 1 + (0.793 + 0.608i)T^{2} \) |
| 5 | \( 1 + (-1.95 - 0.128i)T + (0.991 + 0.130i)T^{2} \) |
| 11 | \( 1 + (0.130 + 0.991i)T^{2} \) |
| 13 | \( 1 + (-1 - i)T + iT^{2} \) |
| 19 | \( 1 + (-0.965 + 0.258i)T^{2} \) |
| 23 | \( 1 + (-0.793 + 0.608i)T^{2} \) |
| 29 | \( 1 + (1.08 - 1.63i)T + (-0.382 - 0.923i)T^{2} \) |
| 31 | \( 1 + (-0.793 - 0.608i)T^{2} \) |
| 37 | \( 1 + (0.293 - 0.257i)T + (0.130 - 0.991i)T^{2} \) |
| 41 | \( 1 + (0.617 + 0.923i)T + (-0.382 + 0.923i)T^{2} \) |
| 43 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 47 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 53 | \( 1 + (-0.607 + 0.465i)T + (0.258 - 0.965i)T^{2} \) |
| 59 | \( 1 + (0.965 + 0.258i)T^{2} \) |
| 61 | \( 1 + (1.49 + 0.735i)T + (0.608 + 0.793i)T^{2} \) |
| 67 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (0.923 - 0.382i)T^{2} \) |
| 73 | \( 1 + (0.172 + 0.349i)T + (-0.608 + 0.793i)T^{2} \) |
| 79 | \( 1 + (0.793 - 0.608i)T^{2} \) |
| 83 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 89 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 97 | \( 1 + (0.324 + 0.216i)T + (0.382 + 0.923i)T^{2} \) |
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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
−8.864590376622901219984830251677, −8.558268995222880104460144118765, −7.11680828152608802901481449440, −6.20621327740264816639306456245, −5.70441412948477036120010817661, −4.97096057954362490057547695658, −3.70311942910689999165918204177, −3.05950550117478176579592925807, −1.92084045888218602138750094691, −1.42732585888155369331131663231,
1.12261177834117106416987761777, 2.33277478868417303839109818425, 3.33131985094536101796315882045, 4.72375564619479817925874630237, 5.48652172124850867912264193607, 5.83241454292286087474848917401, 6.37512413257768213737153324069, 7.45778466342472098859401598981, 8.156999774928680805506422246630, 8.911771657558250029026827354819