L(s) = 1 | + (17.9 + 17.9i)7-s + 4.24i·11-s + (17.9 − 17.9i)13-s + (−76.0 + 76.0i)17-s + 26i·19-s + (76.0 + 76.0i)23-s − 110.·29-s + 52·31-s + (17.9 + 17.9i)37-s + 199. i·41-s + (304. − 304. i)47-s + 299i·49-s + (−380. − 380. i)53-s − 717.·59-s + 350·61-s + ⋯ |
L(s) = 1 | + (0.967 + 0.967i)7-s + 0.116i·11-s + (0.382 − 0.382i)13-s + (−1.08 + 1.08i)17-s + 0.313i·19-s + (0.689 + 0.689i)23-s − 0.706·29-s + 0.301·31-s + (0.0796 + 0.0796i)37-s + 0.759i·41-s + (0.943 − 0.943i)47-s + 0.871i·49-s + (−0.985 − 0.985i)53-s − 1.58·59-s + 0.734·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.391 - 0.920i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.391 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.712005438\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.712005438\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (-17.9 - 17.9i)T + 343iT^{2} \) |
| 11 | \( 1 - 4.24iT - 1.33e3T^{2} \) |
| 13 | \( 1 + (-17.9 + 17.9i)T - 2.19e3iT^{2} \) |
| 17 | \( 1 + (76.0 - 76.0i)T - 4.91e3iT^{2} \) |
| 19 | \( 1 - 26iT - 6.85e3T^{2} \) |
| 23 | \( 1 + (-76.0 - 76.0i)T + 1.21e4iT^{2} \) |
| 29 | \( 1 + 110.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 52T + 2.97e4T^{2} \) |
| 37 | \( 1 + (-17.9 - 17.9i)T + 5.06e4iT^{2} \) |
| 41 | \( 1 - 199. iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 7.95e4iT^{2} \) |
| 47 | \( 1 + (-304. + 304. i)T - 1.03e5iT^{2} \) |
| 53 | \( 1 + (380. + 380. i)T + 1.48e5iT^{2} \) |
| 59 | \( 1 + 717.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 350T + 2.26e5T^{2} \) |
| 67 | \( 1 + (465. + 465. i)T + 3.00e5iT^{2} \) |
| 71 | \( 1 - 517. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (465. - 465. i)T - 3.89e5iT^{2} \) |
| 79 | \( 1 - 1.00e3iT - 4.93e5T^{2} \) |
| 83 | \( 1 + (-456. - 456. i)T + 5.71e5iT^{2} \) |
| 89 | \( 1 + 1.42e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-788. - 788. i)T + 9.12e5iT^{2} \) |
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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
−9.977824204088877659702076410860, −8.949684063285844962035192186424, −8.416523804670263898731189583431, −7.60572105919273145282675175952, −6.43948454780005516299436918066, −5.60612396181705822255862773099, −4.77735364455878318857145666913, −3.67110646621322375183500072352, −2.35878770864439625319660504768, −1.41564813828505974802265384189,
0.44033033436168353163273811267, 1.63498952449543375702359002630, 2.92375853161049807339082684982, 4.31455199938129177085022465117, 4.74496149448753955831523340238, 6.03741488440787233711488737415, 7.08341253752132258743336921499, 7.61092701755609505904913507472, 8.729346666313162074625433995643, 9.297087948179041398136025724514