| L(s) = 1 | − 0.550i·5-s − 4.37·7-s − 2.79i·11-s − i·13-s − 4.67·17-s − 4.05i·19-s − 1.34·23-s + 4.69·25-s − 7.77i·29-s + 8.12·31-s + 2.40i·35-s + 9.06i·37-s − 6.32·41-s + 6.38i·43-s − 2.92·47-s + ⋯ |
| L(s) = 1 | − 0.246i·5-s − 1.65·7-s − 0.843i·11-s − 0.277i·13-s − 1.13·17-s − 0.929i·19-s − 0.279·23-s + 0.939·25-s − 1.44i·29-s + 1.45·31-s + 0.407i·35-s + 1.49i·37-s − 0.987·41-s + 0.974i·43-s − 0.427·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.293 - 0.956i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.293 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3501939777\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3501939777\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 + iT \) |
| good | 5 | \( 1 + 0.550iT - 5T^{2} \) |
| 7 | \( 1 + 4.37T + 7T^{2} \) |
| 11 | \( 1 + 2.79iT - 11T^{2} \) |
| 17 | \( 1 + 4.67T + 17T^{2} \) |
| 19 | \( 1 + 4.05iT - 19T^{2} \) |
| 23 | \( 1 + 1.34T + 23T^{2} \) |
| 29 | \( 1 + 7.77iT - 29T^{2} \) |
| 31 | \( 1 - 8.12T + 31T^{2} \) |
| 37 | \( 1 - 9.06iT - 37T^{2} \) |
| 41 | \( 1 + 6.32T + 41T^{2} \) |
| 43 | \( 1 - 6.38iT - 43T^{2} \) |
| 47 | \( 1 + 2.92T + 47T^{2} \) |
| 53 | \( 1 - 8.02iT - 53T^{2} \) |
| 59 | \( 1 - 2.05iT - 59T^{2} \) |
| 61 | \( 1 + 3.20iT - 61T^{2} \) |
| 67 | \( 1 - 12.5iT - 67T^{2} \) |
| 71 | \( 1 + 5.81T + 71T^{2} \) |
| 73 | \( 1 + 10.4T + 73T^{2} \) |
| 79 | \( 1 - 0.975T + 79T^{2} \) |
| 83 | \( 1 + 1.58iT - 83T^{2} \) |
| 89 | \( 1 - 7.89T + 89T^{2} \) |
| 97 | \( 1 + 13.8T + 97T^{2} \) |
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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
−8.718207208559199447180382610639, −8.189679821360068688111880142348, −7.05835290282729362288627058031, −6.41362082995193951974064115415, −6.04158404836762347583764424104, −4.88404399061240998906741821278, −4.14667619462794606137472901983, −3.03655399899760672380505067798, −2.66824606740002987257526094043, −0.941046688140224049672335337923,
0.12105249206826448109086267598, 1.78504280076139053518374969828, 2.78857197982100312252569651485, 3.56061796884663600846883301090, 4.37212172104051849035801234055, 5.32412778736970864743004212964, 6.32192954842140356961431548394, 6.76376555236088380787944890064, 7.29649277364434132457052445631, 8.439275343924883750267930408258
