| L(s) = 1 | + 1.13e4·3-s − 6.84e5·5-s − 1.14e7·7-s + 4.36e5·9-s − 6.71e8·11-s − 5.24e9·13-s − 7.79e9·15-s + 1.10e10·17-s + 1.02e11·19-s − 1.30e11·21-s − 4.80e11·23-s − 2.94e11·25-s − 1.46e12·27-s + 2.77e12·29-s − 1.01e12·31-s − 7.64e12·33-s + 7.82e12·35-s + 4.01e13·37-s − 5.96e13·39-s + 5.54e13·41-s + 7.56e13·43-s − 2.99e11·45-s + 9.85e13·47-s − 1.01e14·49-s + 1.25e14·51-s + 6.37e14·53-s + 4.59e14·55-s + ⋯ |
| L(s) = 1 | + 1.00·3-s − 0.783·5-s − 0.749·7-s + 0.00338·9-s − 0.944·11-s − 1.78·13-s − 0.785·15-s + 0.382·17-s + 1.38·19-s − 0.750·21-s − 1.27·23-s − 0.385·25-s − 0.998·27-s + 1.02·29-s − 0.214·31-s − 0.945·33-s + 0.587·35-s + 1.87·37-s − 1.78·39-s + 1.08·41-s + 0.987·43-s − 0.00265·45-s + 0.603·47-s − 0.438·49-s + 0.383·51-s + 1.40·53-s + 0.740·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(\approx\) |
\(1.331583479\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.331583479\) |
| \(L(\frac{19}{2})\) |
|
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 | \( 1 - 1.13e4T + 1.29e8T^{2} \) |
| 5 | \( 1 + 6.84e5T + 7.62e11T^{2} \) |
| 7 | \( 1 + 1.14e7T + 2.32e14T^{2} \) |
| 11 | \( 1 + 6.71e8T + 5.05e17T^{2} \) |
| 13 | \( 1 + 5.24e9T + 8.65e18T^{2} \) |
| 17 | \( 1 - 1.10e10T + 8.27e20T^{2} \) |
| 19 | \( 1 - 1.02e11T + 5.48e21T^{2} \) |
| 23 | \( 1 + 4.80e11T + 1.41e23T^{2} \) |
| 29 | \( 1 - 2.77e12T + 7.25e24T^{2} \) |
| 31 | \( 1 + 1.01e12T + 2.25e25T^{2} \) |
| 37 | \( 1 - 4.01e13T + 4.56e26T^{2} \) |
| 41 | \( 1 - 5.54e13T + 2.61e27T^{2} \) |
| 43 | \( 1 - 7.56e13T + 5.87e27T^{2} \) |
| 47 | \( 1 - 9.85e13T + 2.66e28T^{2} \) |
| 53 | \( 1 - 6.37e14T + 2.05e29T^{2} \) |
| 59 | \( 1 + 1.65e15T + 1.27e30T^{2} \) |
| 61 | \( 1 + 2.22e14T + 2.24e30T^{2} \) |
| 67 | \( 1 - 1.72e15T + 1.10e31T^{2} \) |
| 71 | \( 1 + 1.21e15T + 2.96e31T^{2} \) |
| 73 | \( 1 + 9.87e15T + 4.74e31T^{2} \) |
| 79 | \( 1 + 2.90e15T + 1.81e32T^{2} \) |
| 83 | \( 1 - 1.24e16T + 4.21e32T^{2} \) |
| 89 | \( 1 + 2.06e16T + 1.37e33T^{2} \) |
| 97 | \( 1 - 5.68e16T + 5.95e33T^{2} \) |
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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
−11.71324721621998831085627789084, −10.08813695512426906645587389970, −9.354920693723596170910039823561, −7.85004252567785374242735155623, −7.52951412766968965912016892462, −5.71839849982884383208210070087, −4.29645425953651751530204138638, −3.04764494025124927513606055304, −2.43575830526262165590589049456, −0.48096458558087788088470125466,
0.48096458558087788088470125466, 2.43575830526262165590589049456, 3.04764494025124927513606055304, 4.29645425953651751530204138638, 5.71839849982884383208210070087, 7.52951412766968965912016892462, 7.85004252567785374242735155623, 9.354920693723596170910039823561, 10.08813695512426906645587389970, 11.71324721621998831085627789084
