L(s) = 1 | + (−1.32 − 1.80i)5-s − i·7-s + 2.48·11-s − 4.15i·13-s + 5.76i·17-s − 1.60·19-s + 7.28i·23-s + (−1.51 + 4.76i)25-s + 1.45·29-s + 2.24·31-s + (−1.80 + 1.32i)35-s + 6i·37-s − 11.2·41-s + 5.28i·43-s − 3.45i·47-s + ⋯ |
L(s) = 1 | + (−0.590 − 0.807i)5-s − 0.377i·7-s + 0.749·11-s − 1.15i·13-s + 1.39i·17-s − 0.369·19-s + 1.51i·23-s + (−0.303 + 0.952i)25-s + 0.270·29-s + 0.404·31-s + (−0.305 + 0.223i)35-s + 0.986i·37-s − 1.76·41-s + 0.805i·43-s − 0.503i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.807 - 0.590i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.807 - 0.590i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.372396997\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.372396997\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.32 + 1.80i)T \) |
| 7 | \( 1 + iT \) |
good | 11 | \( 1 - 2.48T + 11T^{2} \) |
| 13 | \( 1 + 4.15iT - 13T^{2} \) |
| 17 | \( 1 - 5.76iT - 17T^{2} \) |
| 19 | \( 1 + 1.60T + 19T^{2} \) |
| 23 | \( 1 - 7.28iT - 23T^{2} \) |
| 29 | \( 1 - 1.45T + 29T^{2} \) |
| 31 | \( 1 - 2.24T + 31T^{2} \) |
| 37 | \( 1 - 6iT - 37T^{2} \) |
| 41 | \( 1 + 11.2T + 41T^{2} \) |
| 43 | \( 1 - 5.28iT - 43T^{2} \) |
| 47 | \( 1 + 3.45iT - 47T^{2} \) |
| 53 | \( 1 - 9.21iT - 53T^{2} \) |
| 59 | \( 1 - 5.92T + 59T^{2} \) |
| 61 | \( 1 - 5.35T + 61T^{2} \) |
| 67 | \( 1 + 7.52iT - 67T^{2} \) |
| 71 | \( 1 + 4.24T + 71T^{2} \) |
| 73 | \( 1 - 7.28iT - 73T^{2} \) |
| 79 | \( 1 - 16.9T + 79T^{2} \) |
| 83 | \( 1 + 10.1iT - 83T^{2} \) |
| 89 | \( 1 + 11.4T + 89T^{2} \) |
| 97 | \( 1 - 2.73iT - 97T^{2} \) |
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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
−8.234367645892527957415668269820, −7.80663481686557757030361946138, −6.90751865236674197771857763760, −6.11203567441479992836520005323, −5.35226756800684785171380184659, −4.58280949816514284692313468150, −3.76860237057572529255416818438, −3.25192453055287882380888737702, −1.72698415759832084652201745141, −0.949453857206305336874816334447,
0.44717648594909833894071467562, 1.99818945507953992897763840524, 2.72341613734702548527403455845, 3.68704531189072962434441378059, 4.37611057289622392970141970111, 5.13666978805232637204415158366, 6.27206694770850541121268382110, 6.81481568149221748205866822040, 7.18105554071161069204960286953, 8.300645429606775152210538668619