| L(s) = 1 | − 2.90e3i·3-s − 2.08e4·5-s − 8.01e6i·7-s + 3.45e7·9-s + 1.22e8i·11-s − 4.11e8·13-s + 6.07e7i·15-s + 1.07e10·17-s + 3.07e10i·19-s − 2.33e10·21-s + 6.36e10i·23-s − 1.52e11·25-s − 2.25e11i·27-s + 3.76e11·29-s − 3.36e11i·31-s + ⋯ |
| L(s) = 1 | − 0.443i·3-s − 0.0534·5-s − 1.39i·7-s + 0.803·9-s + 0.572i·11-s − 0.504·13-s + 0.0236i·15-s + 1.54·17-s + 1.81i·19-s − 0.616·21-s + 0.813i·23-s − 0.997·25-s − 0.799i·27-s + 0.751·29-s − 0.394i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(17-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+8) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{17}{2})\) |
\(\approx\) |
\(2.278181090\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.278181090\) |
| \(L(9)\) |
|
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 | \( 1 + 2.90e3iT - 4.30e7T^{2} \) |
| 5 | \( 1 + 2.08e4T + 1.52e11T^{2} \) |
| 7 | \( 1 + 8.01e6iT - 3.32e13T^{2} \) |
| 11 | \( 1 - 1.22e8iT - 4.59e16T^{2} \) |
| 13 | \( 1 + 4.11e8T + 6.65e17T^{2} \) |
| 17 | \( 1 - 1.07e10T + 4.86e19T^{2} \) |
| 19 | \( 1 - 3.07e10iT - 2.88e20T^{2} \) |
| 23 | \( 1 - 6.36e10iT - 6.13e21T^{2} \) |
| 29 | \( 1 - 3.76e11T + 2.50e23T^{2} \) |
| 31 | \( 1 + 3.36e11iT - 7.27e23T^{2} \) |
| 37 | \( 1 + 1.97e12T + 1.23e25T^{2} \) |
| 41 | \( 1 - 4.08e12T + 6.37e25T^{2} \) |
| 43 | \( 1 - 1.53e13iT - 1.36e26T^{2} \) |
| 47 | \( 1 - 2.29e13iT - 5.66e26T^{2} \) |
| 53 | \( 1 - 5.35e13T + 3.87e27T^{2} \) |
| 59 | \( 1 - 6.70e13iT - 2.15e28T^{2} \) |
| 61 | \( 1 + 1.86e14T + 3.67e28T^{2} \) |
| 67 | \( 1 + 1.58e14iT - 1.64e29T^{2} \) |
| 71 | \( 1 - 8.90e14iT - 4.16e29T^{2} \) |
| 73 | \( 1 + 2.83e14T + 6.50e29T^{2} \) |
| 79 | \( 1 + 1.41e15iT - 2.30e30T^{2} \) |
| 83 | \( 1 + 2.82e15iT - 5.07e30T^{2} \) |
| 89 | \( 1 - 2.93e15T + 1.54e31T^{2} \) |
| 97 | \( 1 - 1.34e15T + 6.14e31T^{2} \) |
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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
−11.88425972102710840415784883447, −10.24083023473329554561734163390, −9.862874055945895220708539828609, −7.69645122106972838590829886317, −7.49264314225246397288214855046, −6.01750680067997287864034276345, −4.46367729688238093154556507568, −3.50984360075372047568100391749, −1.70188145466167556440732561058, −0.927216199697044053014307205376,
0.61079088783607264446611136672, 2.17166736023070128345529133542, 3.28471087584462781430301684455, 4.75850099169081720396694155531, 5.69604723508473113958148657892, 7.09058874894666446760653467331, 8.485386627526792805881351983160, 9.425517385599274409535413028674, 10.46107214196602400033872415405, 11.80257145342231179082009891080
