L(s) = 1 | + 2.62e3i·5-s − 2.25e4i·7-s + 2.46e5·11-s − 2.23e5i·13-s + 2.63e6·17-s + 1.10e6·19-s − 7.90e6i·23-s + 2.89e6·25-s + 2.11e6i·29-s + 5.09e7i·31-s + 5.90e7·35-s − 1.66e7i·37-s − 1.11e8·41-s + 6.57e7·43-s − 1.69e8i·47-s + ⋯ |
L(s) = 1 | + 0.839i·5-s − 1.33i·7-s + 1.53·11-s − 0.601i·13-s + 1.85·17-s + 0.447·19-s − 1.22i·23-s + 0.296·25-s + 0.103i·29-s + 1.77i·31-s + 1.12·35-s − 0.240i·37-s − 0.961·41-s + 0.447·43-s − 0.739i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.730 + 0.683i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.730 + 0.683i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{11}{2})\) |
\(\approx\) |
\(2.941760066\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.941760066\) |
\(L(6)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 2.62e3iT - 9.76e6T^{2} \) |
| 7 | \( 1 + 2.25e4iT - 2.82e8T^{2} \) |
| 11 | \( 1 - 2.46e5T + 2.59e10T^{2} \) |
| 13 | \( 1 + 2.23e5iT - 1.37e11T^{2} \) |
| 17 | \( 1 - 2.63e6T + 2.01e12T^{2} \) |
| 19 | \( 1 - 1.10e6T + 6.13e12T^{2} \) |
| 23 | \( 1 + 7.90e6iT - 4.14e13T^{2} \) |
| 29 | \( 1 - 2.11e6iT - 4.20e14T^{2} \) |
| 31 | \( 1 - 5.09e7iT - 8.19e14T^{2} \) |
| 37 | \( 1 + 1.66e7iT - 4.80e15T^{2} \) |
| 41 | \( 1 + 1.11e8T + 1.34e16T^{2} \) |
| 43 | \( 1 - 6.57e7T + 2.16e16T^{2} \) |
| 47 | \( 1 + 1.69e8iT - 5.25e16T^{2} \) |
| 53 | \( 1 - 8.54e7iT - 1.74e17T^{2} \) |
| 59 | \( 1 + 3.55e8T + 5.11e17T^{2} \) |
| 61 | \( 1 + 9.81e8iT - 7.13e17T^{2} \) |
| 67 | \( 1 + 7.02e8T + 1.82e18T^{2} \) |
| 71 | \( 1 + 2.41e9iT - 3.25e18T^{2} \) |
| 73 | \( 1 + 2.80e9T + 4.29e18T^{2} \) |
| 79 | \( 1 + 1.18e9iT - 9.46e18T^{2} \) |
| 83 | \( 1 - 6.20e9T + 1.55e19T^{2} \) |
| 89 | \( 1 - 2.60e9T + 3.11e19T^{2} \) |
| 97 | \( 1 + 7.03e9T + 7.37e19T^{2} \) |
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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
−10.34159905844400113170834720134, −9.089397650899158529227484381772, −7.84850814245202410085758695353, −7.02827214792827768319636038785, −6.33901623578860343528222266228, −4.94948606798201538104238119407, −3.68803545135603561303863438182, −3.14960959196841022372317164443, −1.42040614740454701485074472572, −0.66823567353792597447771636783,
0.988052904305749140574126190490, 1.72384853946861949395283342285, 3.13114559571592966045213691807, 4.23380452783987489240094028798, 5.43269930791586275154501967067, 6.04362283527909445450765277857, 7.39345465267381867727153177362, 8.487326863082824285962611712240, 9.278434155024739125563548049595, 9.764616033706243712467540069213