| L(s) = 1 | − 1.13e5i·3-s + 2.14e7i·5-s + 5.47e8·7-s − 2.31e9·9-s − 1.01e11i·11-s + 4.95e11i·13-s + 2.42e12·15-s − 1.25e13·17-s − 2.73e12i·19-s − 6.19e13i·21-s + 1.32e14·23-s + 1.81e13·25-s − 9.20e14i·27-s + 2.72e15i·29-s − 4.42e15·31-s + ⋯ |
| L(s) = 1 | − 1.10i·3-s + 0.980i·5-s + 0.733·7-s − 0.221·9-s − 1.17i·11-s + 0.997i·13-s + 1.08·15-s − 1.50·17-s − 0.102i·19-s − 0.810i·21-s + 0.666·23-s + 0.0379·25-s − 0.860i·27-s + 1.20i·29-s − 0.969·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 + 0.258i)\, \overline{\Lambda}(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (-0.965 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(\approx\) |
\(0.9673518881\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9673518881\) |
| \(L(\frac{23}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + 1.13e5iT - 1.04e10T^{2} \) |
| 5 | \( 1 - 2.14e7iT - 4.76e14T^{2} \) |
| 7 | \( 1 - 5.47e8T + 5.58e17T^{2} \) |
| 11 | \( 1 + 1.01e11iT - 7.40e21T^{2} \) |
| 13 | \( 1 - 4.95e11iT - 2.47e23T^{2} \) |
| 17 | \( 1 + 1.25e13T + 6.90e25T^{2} \) |
| 19 | \( 1 + 2.73e12iT - 7.14e26T^{2} \) |
| 23 | \( 1 - 1.32e14T + 3.94e28T^{2} \) |
| 29 | \( 1 - 2.72e15iT - 5.13e30T^{2} \) |
| 31 | \( 1 + 4.42e15T + 2.08e31T^{2} \) |
| 37 | \( 1 - 8.33e15iT - 8.55e32T^{2} \) |
| 41 | \( 1 + 7.49e16T + 7.38e33T^{2} \) |
| 43 | \( 1 - 2.77e16iT - 2.00e34T^{2} \) |
| 47 | \( 1 - 1.56e17T + 1.30e35T^{2} \) |
| 53 | \( 1 - 6.67e17iT - 1.62e36T^{2} \) |
| 59 | \( 1 + 5.80e18iT - 1.54e37T^{2} \) |
| 61 | \( 1 + 9.87e18iT - 3.10e37T^{2} \) |
| 67 | \( 1 + 2.22e19iT - 2.22e38T^{2} \) |
| 71 | \( 1 - 2.93e19T + 7.52e38T^{2} \) |
| 73 | \( 1 - 3.26e19T + 1.34e39T^{2} \) |
| 79 | \( 1 - 5.86e19T + 7.08e39T^{2} \) |
| 83 | \( 1 + 9.06e19iT - 1.99e40T^{2} \) |
| 89 | \( 1 + 1.01e20T + 8.65e40T^{2} \) |
| 97 | \( 1 + 6.59e20T + 5.27e41T^{2} \) |
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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
−10.94468983259549684660585950153, −9.116622457563670619251950720576, −8.083296306774360966278013572930, −6.88746446024281040778482466274, −6.49360238893375122643822586112, −4.91166095170117254356841013843, −3.44794334429954827243511263061, −2.24332134540101754634439616101, −1.44496783955737445160195840115, −0.17716790014847852230648570861,
1.15294111241244790588705485594, 2.33483454070677454363799592718, 3.98571289291837584285287196649, 4.69068172339541808389559504035, 5.38535702962619433215113029018, 7.14833733961798601026813696809, 8.423346754903769966544255475019, 9.284747445021625846570699665605, 10.26174073628667053972962827968, 11.22078657440691315085563812961
