| L(s) = 1 | − 4.82e4·5-s − 9.88e5i·7-s − 2.32e7i·11-s − 1.22e8·13-s − 5.67e8·17-s + 1.44e9i·19-s + 4.82e9i·23-s − 3.77e9·25-s + 2.25e10·29-s + 9.74e9i·31-s + 4.76e10i·35-s − 4.31e10·37-s − 2.29e11·41-s + 1.08e11i·43-s − 1.42e11i·47-s + ⋯ |
| L(s) = 1 | − 0.617·5-s − 1.19i·7-s − 1.19i·11-s − 1.95·13-s − 1.38·17-s + 1.61i·19-s + 1.41i·23-s − 0.618·25-s + 1.30·29-s + 0.354i·31-s + 0.741i·35-s − 0.454·37-s − 1.18·41-s + 0.399i·43-s − 0.281i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{15}{2})\) |
\(\approx\) |
\(0.6633045321\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6633045321\) |
| \(L(8)\) |
|
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 | \( 1 + 4.82e4T + 6.10e9T^{2} \) |
| 7 | \( 1 + 9.88e5iT - 6.78e11T^{2} \) |
| 11 | \( 1 + 2.32e7iT - 3.79e14T^{2} \) |
| 13 | \( 1 + 1.22e8T + 3.93e15T^{2} \) |
| 17 | \( 1 + 5.67e8T + 1.68e17T^{2} \) |
| 19 | \( 1 - 1.44e9iT - 7.99e17T^{2} \) |
| 23 | \( 1 - 4.82e9iT - 1.15e19T^{2} \) |
| 29 | \( 1 - 2.25e10T + 2.97e20T^{2} \) |
| 31 | \( 1 - 9.74e9iT - 7.56e20T^{2} \) |
| 37 | \( 1 + 4.31e10T + 9.01e21T^{2} \) |
| 41 | \( 1 + 2.29e11T + 3.79e22T^{2} \) |
| 43 | \( 1 - 1.08e11iT - 7.38e22T^{2} \) |
| 47 | \( 1 + 1.42e11iT - 2.56e23T^{2} \) |
| 53 | \( 1 + 5.07e11T + 1.37e24T^{2} \) |
| 59 | \( 1 + 4.16e12iT - 6.19e24T^{2} \) |
| 61 | \( 1 + 1.20e12T + 9.87e24T^{2} \) |
| 67 | \( 1 - 3.78e12iT - 3.67e25T^{2} \) |
| 71 | \( 1 + 1.50e13iT - 8.27e25T^{2} \) |
| 73 | \( 1 - 1.11e12T + 1.22e26T^{2} \) |
| 79 | \( 1 - 2.64e13iT - 3.68e26T^{2} \) |
| 83 | \( 1 - 5.32e12iT - 7.36e26T^{2} \) |
| 89 | \( 1 - 6.78e13T + 1.95e27T^{2} \) |
| 97 | \( 1 - 3.20e13T + 6.52e27T^{2} \) |
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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
−10.50909518930002331623586962778, −9.664772672159505722815280952515, −8.261625256203167991471950772750, −7.52095658897517708772323822498, −6.55211628103735740645876055959, −5.13545229158821430082198511609, −4.07952167583681283127960922745, −3.19770494084161949665296299516, −1.73062646227418172965585672390, −0.40689803543843700321622982028,
0.27126732748732008471534132764, 2.23668408757563821392417483528, 2.59749207652868943835624900427, 4.51458898450864199040733853139, 4.92971958073303904263190498877, 6.57669609244424121108237451967, 7.34772706733674192064643441042, 8.596422975496562207633182075964, 9.400864167198789121520683642402, 10.47309854633124878970612402160
