| L(s) = 1 | − 9.74i·2-s + 1.95e3·4-s − 1.03e4i·5-s − 8.66e3i·7-s − 3.89e4i·8-s − 1.00e5·10-s + 1.65e5i·11-s + (−6.11e5 + 1.19e6i)13-s − 8.44e4·14-s + 3.61e6·16-s + 9.62e6·17-s − 1.70e7i·19-s − 2.01e7i·20-s + 1.61e6·22-s + 2.24e7·23-s + ⋯ |
| L(s) = 1 | − 0.215i·2-s + 0.953·4-s − 1.47i·5-s − 0.194i·7-s − 0.420i·8-s − 0.317·10-s + 0.309i·11-s + (−0.456 + 0.889i)13-s − 0.0419·14-s + 0.862·16-s + 1.64·17-s − 1.57i·19-s − 1.40i·20-s + 0.0666·22-s + 0.726·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.456 + 0.889i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.456 + 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(2.940311615\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.940311615\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 + (6.11e5 - 1.19e6i)T \) |
| good | 2 | \( 1 + 9.74iT - 2.04e3T^{2} \) |
| 5 | \( 1 + 1.03e4iT - 4.88e7T^{2} \) |
| 7 | \( 1 + 8.66e3iT - 1.97e9T^{2} \) |
| 11 | \( 1 - 1.65e5iT - 2.85e11T^{2} \) |
| 17 | \( 1 - 9.62e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + 1.70e7iT - 1.16e14T^{2} \) |
| 23 | \( 1 - 2.24e7T + 9.52e14T^{2} \) |
| 29 | \( 1 - 1.47e8T + 1.22e16T^{2} \) |
| 31 | \( 1 + 9.15e7iT - 2.54e16T^{2} \) |
| 37 | \( 1 + 3.15e8iT - 1.77e17T^{2} \) |
| 41 | \( 1 - 2.02e7iT - 5.50e17T^{2} \) |
| 43 | \( 1 + 3.31e8T + 9.29e17T^{2} \) |
| 47 | \( 1 - 4.54e8iT - 2.47e18T^{2} \) |
| 53 | \( 1 + 3.55e9T + 9.26e18T^{2} \) |
| 59 | \( 1 + 3.86e9iT - 3.01e19T^{2} \) |
| 61 | \( 1 + 4.27e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 8.03e9iT - 1.22e20T^{2} \) |
| 71 | \( 1 + 1.87e10iT - 2.31e20T^{2} \) |
| 73 | \( 1 - 5.36e9iT - 3.13e20T^{2} \) |
| 79 | \( 1 + 4.48e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 2.99e10iT - 1.28e21T^{2} \) |
| 89 | \( 1 - 6.53e10iT - 2.77e21T^{2} \) |
| 97 | \( 1 + 1.14e11iT - 7.15e21T^{2} \) |
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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
−11.23101385114392928704119415938, −9.932458761975771004386263901284, −9.039424805067686453161656626712, −7.78709467591712778104737039130, −6.77490083988916160410638701187, −5.36691538758885287929217465861, −4.37369640280906286909303309067, −2.82166124882454064885466158465, −1.50728726285768694954926247405, −0.66139090889596746294508724203,
1.32307668505625585069540084070, 2.80688698084376506222223537017, 3.30028401127662422149150664512, 5.48954459560532251618899326601, 6.34933233658312101535452170245, 7.37398038304196304078869020751, 8.133198887230438390633262752293, 10.16851182724414798783012065558, 10.46419571877854420773712289291, 11.69314226466454962647407893091
