L(s) = 1 | + 8.57i·3-s + (−0.582 + 11.1i)5-s + 22.1i·7-s − 46.4·9-s + 27.1·11-s − 70.3i·13-s + (−95.7 − 4.99i)15-s + 73.3i·17-s + 110.·19-s − 189.·21-s − 107. i·23-s + (−124. − 13.0i)25-s − 167. i·27-s − 68.6·29-s − 137.·31-s + ⋯ |
L(s) = 1 | + 1.64i·3-s + (−0.0521 + 0.998i)5-s + 1.19i·7-s − 1.72·9-s + 0.743·11-s − 1.50i·13-s + (−1.64 − 0.0859i)15-s + 1.04i·17-s + 1.32·19-s − 1.97·21-s − 0.977i·23-s + (−0.994 − 0.104i)25-s − 1.19i·27-s − 0.439·29-s − 0.794·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0521i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0521i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.0388225 + 1.48909i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0388225 + 1.48909i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.582 - 11.1i)T \) |
good | 3 | \( 1 - 8.57iT - 27T^{2} \) |
| 7 | \( 1 - 22.1iT - 343T^{2} \) |
| 11 | \( 1 - 27.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 70.3iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 73.3iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 110.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 107. iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 68.6T + 2.43e4T^{2} \) |
| 31 | \( 1 + 137.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 60.3iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 95.1T + 6.89e4T^{2} \) |
| 43 | \( 1 - 501. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 439. iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 286. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 547.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 511.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 301. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 82.8T + 3.57e5T^{2} \) |
| 73 | \( 1 + 763. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 1.01e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 704. iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 743.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.13e3iT - 9.12e5T^{2} \) |
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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
−12.78431922799145402297042473433, −11.57613117909130838297339748789, −10.78947480666444603957708240577, −9.940053411877276118572797947245, −9.111113832517079999205055208127, −7.955809438064826697309839414213, −6.15327562798036571222126586331, −5.29429227994771338051606049269, −3.76621696162690482758448157665, −2.81761714331742832738234726732,
0.74177337154159347003225054451, 1.72246242498007187356295720300, 3.92580112744876274726015212916, 5.48508624881498176015951399284, 7.07900258856973387248840941103, 7.28899835514460645325450007568, 8.714249967145629472395994164130, 9.640626254618253328806902630376, 11.61311354526994909941417169089, 11.79468768450780332416873524534