| L(s) = 1 | − i·2-s − 4-s + i·5-s − i·7-s + i·8-s + 10-s − 3i·11-s + (3 − 2i)13-s − 14-s + 16-s − 7·17-s + 3i·19-s − i·20-s − 3·22-s − 23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s + 0.447i·5-s − 0.377i·7-s + 0.353i·8-s + 0.316·10-s − 0.904i·11-s + (0.832 − 0.554i)13-s − 0.267·14-s + 0.250·16-s − 1.69·17-s + 0.688i·19-s − 0.223i·20-s − 0.639·22-s − 0.208·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.104482834\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.104482834\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 + iT \) |
| 13 | \( 1 + (-3 + 2i)T \) |
| good | 5 | \( 1 - iT - 5T^{2} \) |
| 11 | \( 1 + 3iT - 11T^{2} \) |
| 17 | \( 1 + 7T + 17T^{2} \) |
| 19 | \( 1 - 3iT - 19T^{2} \) |
| 23 | \( 1 + T + 23T^{2} \) |
| 29 | \( 1 - T + 29T^{2} \) |
| 31 | \( 1 + 8iT - 31T^{2} \) |
| 37 | \( 1 + iT - 37T^{2} \) |
| 41 | \( 1 + 4iT - 41T^{2} \) |
| 43 | \( 1 + 5T + 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + 10iT - 59T^{2} \) |
| 61 | \( 1 + 13T + 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 + 6iT - 71T^{2} \) |
| 73 | \( 1 + 13iT - 73T^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 + 2iT - 83T^{2} \) |
| 89 | \( 1 + 12iT - 89T^{2} \) |
| 97 | \( 1 - 6iT - 97T^{2} \) |
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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
−9.036663169766964913227074125002, −8.447212871770216950398191208592, −7.60679165180694987201629707348, −6.49370950599220558137566910917, −5.88599909693640224044930901627, −4.69292064562987705603786843608, −3.78409249506407770586609364440, −3.02937892189993561468092851632, −1.87768797677584806266008985872, −0.43449154123014821579076293092,
1.44992328845520740233996247774, 2.75512917064513905378117279942, 4.22323689852297148441228420542, 4.70997468921991152345028421615, 5.65882027537937429605788620253, 6.77232587042868220330736864610, 6.96367648577919162621794008613, 8.348830182637645133718463353099, 8.773497173681986127034185370986, 9.376410509124303048516779698121
