| L(s) = 1 | − 1.73·3-s + 2i·5-s + 6.92i·7-s + 2.99·9-s + 6.92·11-s + 2i·13-s − 3.46i·15-s + 10·17-s − 20.7·19-s − 11.9i·21-s + 27.7i·23-s + 21·25-s − 5.19·27-s − 26i·29-s + 6.92i·31-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.400i·5-s + 0.989i·7-s + 0.333·9-s + 0.629·11-s + 0.153i·13-s − 0.230i·15-s + 0.588·17-s − 1.09·19-s − 0.571i·21-s + 1.20i·23-s + 0.839·25-s − 0.192·27-s − 0.896i·29-s + 0.223i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.000377238\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.000377238\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.73T \) |
| good | 5 | \( 1 - 2iT - 25T^{2} \) |
| 7 | \( 1 - 6.92iT - 49T^{2} \) |
| 11 | \( 1 - 6.92T + 121T^{2} \) |
| 13 | \( 1 - 2iT - 169T^{2} \) |
| 17 | \( 1 - 10T + 289T^{2} \) |
| 19 | \( 1 + 20.7T + 361T^{2} \) |
| 23 | \( 1 - 27.7iT - 529T^{2} \) |
| 29 | \( 1 + 26iT - 841T^{2} \) |
| 31 | \( 1 - 6.92iT - 961T^{2} \) |
| 37 | \( 1 + 26iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 58T + 1.68e3T^{2} \) |
| 43 | \( 1 + 48.4T + 1.84e3T^{2} \) |
| 47 | \( 1 - 69.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 74iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 90.0T + 3.48e3T^{2} \) |
| 61 | \( 1 - 26iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 6.92T + 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 - 46T + 5.32e3T^{2} \) |
| 79 | \( 1 - 117. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 48.4T + 6.88e3T^{2} \) |
| 89 | \( 1 + 82T + 7.92e3T^{2} \) |
| 97 | \( 1 - 2T + 9.40e3T^{2} \) |
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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
−10.53105049051710671845830660918, −9.577907552187400906535384487205, −8.852482538674464373465295818004, −7.83695347113070613376341695246, −6.77220620602220603836783771863, −6.06526451728543956581084483531, −5.22358418500943377619517409984, −4.07731934889746550170118541388, −2.85167079240293448164867820453, −1.53037724744657528560640122046,
0.38517912918712351262671873339, 1.59585418221718584366218108143, 3.38634949907889964494500267356, 4.42429715335390583686343522269, 5.17260288467470932468080293448, 6.48713204203627230311641901813, 6.93162356643384926375795708832, 8.161391500226184061540209203763, 8.873344296312076914297467242686, 10.11255600465027804613266528822
