L(s) = 1 | − 2.93·5-s − 4.82i·11-s + 2.93i·13-s + 7.07·17-s − 5.86i·19-s + 2i·23-s + 3.58·25-s − 0.828i·29-s + 5.86i·31-s − 5.41·37-s + 1.21·41-s − 4.48·43-s + 5.86·47-s − 7.07i·53-s + 14.1i·55-s + ⋯ |
L(s) = 1 | − 1.31·5-s − 1.45i·11-s + 0.812i·13-s + 1.71·17-s − 1.34i·19-s + 0.417i·23-s + 0.717·25-s − 0.153i·29-s + 1.05i·31-s − 0.890·37-s + 0.189·41-s − 0.683·43-s + 0.854·47-s − 0.971i·53-s + 1.90i·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.192 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.192 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.065664010\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.065664010\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
good | 5 | \( 1 + 2.93T + 5T^{2} \) |
| 11 | \( 1 + 4.82iT - 11T^{2} \) |
| 13 | \( 1 - 2.93iT - 13T^{2} \) |
| 17 | \( 1 - 7.07T + 17T^{2} \) |
| 19 | \( 1 + 5.86iT - 19T^{2} \) |
| 23 | \( 1 - 2iT - 23T^{2} \) |
| 29 | \( 1 + 0.828iT - 29T^{2} \) |
| 31 | \( 1 - 5.86iT - 31T^{2} \) |
| 37 | \( 1 + 5.41T + 37T^{2} \) |
| 41 | \( 1 - 1.21T + 41T^{2} \) |
| 43 | \( 1 + 4.48T + 43T^{2} \) |
| 47 | \( 1 - 5.86T + 47T^{2} \) |
| 53 | \( 1 + 7.07iT - 53T^{2} \) |
| 59 | \( 1 - 5.86T + 59T^{2} \) |
| 61 | \( 1 - 1.21iT - 61T^{2} \) |
| 67 | \( 1 - 8.48T + 67T^{2} \) |
| 71 | \( 1 + 0.828iT - 71T^{2} \) |
| 73 | \( 1 - 7.07iT - 73T^{2} \) |
| 79 | \( 1 + 1.65T + 79T^{2} \) |
| 83 | \( 1 + 11.7T + 83T^{2} \) |
| 89 | \( 1 - 11.2T + 89T^{2} \) |
| 97 | \( 1 + 7.07iT - 97T^{2} \) |
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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
−7.74755663856340993266317330010, −7.13167535500710769324446678200, −6.48904837323152790168524016364, −5.50086298623551835165726038433, −4.96651550661391013181825978669, −3.88560367415907970986955049961, −3.50135839114853317794194900291, −2.72064048450740367774580464262, −1.27567932158929819208969679113, −0.34343656218144123503072271666,
0.936199096923585792646386629966, 2.05120183344966994532669281005, 3.18939853358993259507778788679, 3.78443863753041354592687700290, 4.44726249379117533325626059024, 5.29932510429385708256030943914, 5.94710343463148412496160771474, 7.00733516184432333331616375637, 7.60807728000517134218827097326, 7.909472561937407414031055688498