| L(s) = 1 | − 12.0·5-s + (14.8 − 11.0i)7-s − 0.00255i·11-s + 7.00i·13-s − 28.3·17-s − 49.1i·19-s + 51.3i·23-s + 20.7·25-s − 113. i·29-s − 32.8i·31-s + (−179. + 133. i)35-s − 101.·37-s + 22.4·41-s + 454.·43-s − 462.·47-s + ⋯ |
| L(s) = 1 | − 1.07·5-s + (0.803 − 0.595i)7-s − 7.00e − 5i·11-s + 0.149i·13-s − 0.403·17-s − 0.594i·19-s + 0.465i·23-s + 0.165·25-s − 0.724i·29-s − 0.190i·31-s + (−0.867 + 0.642i)35-s − 0.453·37-s + 0.0856·41-s + 1.61·43-s − 1.43·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.595 - 0.803i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.595 - 0.803i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3683548239\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3683548239\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-14.8 + 11.0i)T \) |
| good | 5 | \( 1 + 12.0T + 125T^{2} \) |
| 11 | \( 1 + 0.00255iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 7.00iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 28.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 49.1iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 51.3iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 113. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 32.8iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 101.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 22.4T + 6.89e4T^{2} \) |
| 43 | \( 1 - 454.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 462.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 567. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 291.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 683. iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 539.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 307. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 495. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 649.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.13e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 130.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.45e3iT - 9.12e5T^{2} \) |
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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
−8.808714651924949846465896263584, −8.019489147169144364525860537201, −7.53204249127460823046663968838, −6.83074709155713397681316227082, −5.77568007116097155214255900536, −4.69679101554220077082651226303, −4.21896027806808773321984742764, −3.34100320626888310855664522827, −2.12723685161971465440774890676, −0.939095863396687137921861944224,
0.087025405467247765805647466572, 1.39646954148055778060451104963, 2.48819450300369159280532911964, 3.54998073238530426093351106913, 4.36245166185117496864362101763, 5.12982011183184782670177534153, 5.99747693648017329558775417092, 7.01106015936657427506786488143, 7.73416936452973420336375204846, 8.387593185829792378315238130255
