L(s) = 1 | + (−0.780 + 1.17i)2-s + (0.848 + 1.51i)3-s + (−0.780 − 1.84i)4-s + (−2.44 − 0.179i)6-s + 3.02·7-s + (2.78 + 0.516i)8-s + (−1.56 + 2.56i)9-s + 1.32·11-s + (2.11 − 2.74i)12-s + 5.12i·13-s + (−2.35 + 3.56i)14-s + (−2.78 + 2.87i)16-s − 2·17-s + (−1.80 − 3.84i)18-s − 1.32i·19-s + ⋯ |
L(s) = 1 | + (−0.552 + 0.833i)2-s + (0.489 + 0.871i)3-s + (−0.390 − 0.920i)4-s + (−0.997 − 0.0731i)6-s + 1.14·7-s + (0.983 + 0.182i)8-s + (−0.520 + 0.853i)9-s + 0.399·11-s + (0.611 − 0.791i)12-s + 1.42i·13-s + (−0.630 + 0.951i)14-s + (−0.695 + 0.718i)16-s − 0.485·17-s + (−0.424 − 0.905i)18-s − 0.303i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.434 - 0.900i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.434 - 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.641217 + 1.02080i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.641217 + 1.02080i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.780 - 1.17i)T \) |
| 3 | \( 1 + (-0.848 - 1.51i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 3.02T + 7T^{2} \) |
| 11 | \( 1 - 1.32T + 11T^{2} \) |
| 13 | \( 1 - 5.12iT - 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + 1.32iT - 19T^{2} \) |
| 23 | \( 1 + 0.371iT - 23T^{2} \) |
| 29 | \( 1 + 3.12iT - 29T^{2} \) |
| 31 | \( 1 + 4.71iT - 31T^{2} \) |
| 37 | \( 1 - 5.12iT - 37T^{2} \) |
| 41 | \( 1 + 1.12iT - 41T^{2} \) |
| 43 | \( 1 - 7.73T + 43T^{2} \) |
| 47 | \( 1 - 3.02iT - 47T^{2} \) |
| 53 | \( 1 + 12.2T + 53T^{2} \) |
| 59 | \( 1 - 14.1T + 59T^{2} \) |
| 61 | \( 1 - 3.12T + 61T^{2} \) |
| 67 | \( 1 - 4.34T + 67T^{2} \) |
| 71 | \( 1 - 3.39T + 71T^{2} \) |
| 73 | \( 1 + 8.24iT - 73T^{2} \) |
| 79 | \( 1 + 8.10iT - 79T^{2} \) |
| 83 | \( 1 + 15.1iT - 83T^{2} \) |
| 89 | \( 1 + 10.2iT - 89T^{2} \) |
| 97 | \( 1 + 6iT - 97T^{2} \) |
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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
−11.54797149844107847223787404178, −11.03271518319318675377464212378, −9.851206890433187231310380488869, −9.096203754647435612933502626495, −8.374196095455567404346437162224, −7.43229283171690868144656732184, −6.19596800236446180978671702234, −4.85384875188578496568772188824, −4.21792540070786031498926910673, −1.97834830941150648391040170243,
1.16597243056715188675866212877, 2.45420318613351866694000462556, 3.75602648697378891801418330507, 5.29781060726282089725950437005, 6.93957139303775724867126164478, 7.960703545267705464542239027860, 8.437996530335463863722850923976, 9.468234400232142016585517264215, 10.69837490167757777119975364073, 11.39249640257107027393871966641