L(s) = 1 | − 0.162·5-s + i·7-s + 2.40i·11-s + 4.76i·13-s + 2.74i·17-s + 1.86·19-s − 1.32·23-s − 4.97·25-s − 3.96·29-s − 8.01i·31-s − 0.162i·35-s + 6.09i·37-s + 3.16i·41-s + 1.70·43-s − 12.1·47-s + ⋯ |
L(s) = 1 | − 0.0724·5-s + 0.377i·7-s + 0.724i·11-s + 1.32i·13-s + 0.665i·17-s + 0.426·19-s − 0.277·23-s − 0.994·25-s − 0.735·29-s − 1.43i·31-s − 0.0273i·35-s + 1.00i·37-s + 0.494i·41-s + 0.260·43-s − 1.77·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.108i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 + 0.108i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5650731520\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5650731520\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - iT \) |
good | 5 | \( 1 + 0.162T + 5T^{2} \) |
| 11 | \( 1 - 2.40iT - 11T^{2} \) |
| 13 | \( 1 - 4.76iT - 13T^{2} \) |
| 17 | \( 1 - 2.74iT - 17T^{2} \) |
| 19 | \( 1 - 1.86T + 19T^{2} \) |
| 23 | \( 1 + 1.32T + 23T^{2} \) |
| 29 | \( 1 + 3.96T + 29T^{2} \) |
| 31 | \( 1 + 8.01iT - 31T^{2} \) |
| 37 | \( 1 - 6.09iT - 37T^{2} \) |
| 41 | \( 1 - 3.16iT - 41T^{2} \) |
| 43 | \( 1 - 1.70T + 43T^{2} \) |
| 47 | \( 1 + 12.1T + 47T^{2} \) |
| 53 | \( 1 - 8.22T + 53T^{2} \) |
| 59 | \( 1 - 5.45iT - 59T^{2} \) |
| 61 | \( 1 - 4.28iT - 61T^{2} \) |
| 67 | \( 1 - 4.61T + 67T^{2} \) |
| 71 | \( 1 + 13.9T + 71T^{2} \) |
| 73 | \( 1 + 6.37T + 73T^{2} \) |
| 79 | \( 1 + 9.75iT - 79T^{2} \) |
| 83 | \( 1 + 16.4iT - 83T^{2} \) |
| 89 | \( 1 + 2.35iT - 89T^{2} \) |
| 97 | \( 1 - 14.4T + 97T^{2} \) |
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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
−8.456657106822548441806062232282, −7.65142297663997960629279508377, −7.12467812198161735987913546695, −6.20438389078143442888373793075, −5.76819025108907843618684544379, −4.64500930872610302849800091837, −4.20565085947978584011092499310, −3.27535600568013267637577697051, −2.15408847156672865822450928861, −1.58445878275771928526532611161,
0.14572212169402424235894592357, 1.15892986908857978587009040857, 2.39346105864594494281275289366, 3.34100687572114034031108300446, 3.81059995086622059276105095622, 5.00340687002863745401813787983, 5.48772629901710437624762131248, 6.23657423966478219592796078047, 7.13524015399172033717695680868, 7.70807826885304009556524918736