L(s) = 1 | + (−3.03 − 4.21i)3-s − 3.22i·5-s + 7i·7-s + (−8.57 + 25.6i)9-s + 9.66·11-s + 34.8·13-s + (−13.5 + 9.78i)15-s + 129. i·17-s − 67.0i·19-s + (29.5 − 21.2i)21-s − 15.6·23-s + 114.·25-s + (133. − 41.5i)27-s + 87.6i·29-s − 143. i·31-s + ⋯ |
L(s) = 1 | + (−0.584 − 0.811i)3-s − 0.288i·5-s + 0.377i·7-s + (−0.317 + 0.948i)9-s + 0.264·11-s + 0.743·13-s + (−0.234 + 0.168i)15-s + 1.84i·17-s − 0.809i·19-s + (0.306 − 0.220i)21-s − 0.141·23-s + 0.916·25-s + (0.955 − 0.296i)27-s + 0.561i·29-s − 0.831i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.911 + 0.410i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.911 + 0.410i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.509259626\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.509259626\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (3.03 + 4.21i)T \) |
| 7 | \( 1 - 7iT \) |
good | 5 | \( 1 + 3.22iT - 125T^{2} \) |
| 11 | \( 1 - 9.66T + 1.33e3T^{2} \) |
| 13 | \( 1 - 34.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 129. iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 67.0iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 15.6T + 1.21e4T^{2} \) |
| 29 | \( 1 - 87.6iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 143. iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 104.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 257. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 267. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 430.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 121. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 861.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 502.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 162. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 616.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 719.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 250. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 376.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 870. iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 44.3T + 9.12e5T^{2} \) |
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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
−11.11546124584573641304446164091, −10.42340383678858418722529756995, −8.916829535372927816848490768595, −8.306670022498282970162008929633, −7.09543358522403159762750876939, −6.17468219723831695102298033275, −5.35581039763763366329298744489, −3.95716424013047901071593748684, −2.22080635324397030991465986052, −0.930263449900895281392887587638,
0.840209307582754356966457892149, 3.00190144109218809359344483540, 4.12590769875754617905880618419, 5.16827104008742931529408840272, 6.26450868742137466049691579690, 7.18997260176997786326446607705, 8.535499461708363239343554644854, 9.546042997529273748739909052959, 10.26314615440981934656465729712, 11.22498927193004532883047922589