| L(s) = 1 | + 128·2-s − 5.81e3·3-s + 1.63e4·4-s + 7.81e4·5-s − 7.44e5·6-s − 2.56e6·7-s + 2.09e6·8-s + 1.94e7·9-s + 1.00e7·10-s + 1.14e8·11-s − 9.52e7·12-s + 2.84e8·13-s − 3.28e8·14-s − 4.54e8·15-s + 2.68e8·16-s + 7.76e8·17-s + 2.49e9·18-s + 3.74e9·19-s + 1.28e9·20-s + 1.49e10·21-s + 1.46e10·22-s − 2.76e10·23-s − 1.21e10·24-s + 6.10e9·25-s + 3.64e10·26-s − 2.98e10·27-s − 4.19e10·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.53·3-s + 0.5·4-s + 0.447·5-s − 1.08·6-s − 1.17·7-s + 0.353·8-s + 1.35·9-s + 0.316·10-s + 1.77·11-s − 0.767·12-s + 1.25·13-s − 0.831·14-s − 0.686·15-s + 0.250·16-s + 0.459·17-s + 0.960·18-s + 0.959·19-s + 0.223·20-s + 1.80·21-s + 1.25·22-s − 1.69·23-s − 0.542·24-s + 0.200·25-s + 0.888·26-s − 0.549·27-s − 0.588·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(\approx\) |
\(1.834104002\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.834104002\) |
| \(L(\frac{17}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 128T \) |
| 5 | \( 1 - 7.81e4T \) |
| good | 3 | \( 1 + 5.81e3T + 1.43e7T^{2} \) |
| 7 | \( 1 + 2.56e6T + 4.74e12T^{2} \) |
| 11 | \( 1 - 1.14e8T + 4.17e15T^{2} \) |
| 13 | \( 1 - 2.84e8T + 5.11e16T^{2} \) |
| 17 | \( 1 - 7.76e8T + 2.86e18T^{2} \) |
| 19 | \( 1 - 3.74e9T + 1.51e19T^{2} \) |
| 23 | \( 1 + 2.76e10T + 2.66e20T^{2} \) |
| 29 | \( 1 - 1.59e10T + 8.62e21T^{2} \) |
| 31 | \( 1 - 2.79e9T + 2.34e22T^{2} \) |
| 37 | \( 1 - 6.85e11T + 3.33e23T^{2} \) |
| 41 | \( 1 - 9.09e11T + 1.55e24T^{2} \) |
| 43 | \( 1 + 4.11e11T + 3.17e24T^{2} \) |
| 47 | \( 1 + 4.29e11T + 1.20e25T^{2} \) |
| 53 | \( 1 - 1.11e13T + 7.31e25T^{2} \) |
| 59 | \( 1 + 2.98e13T + 3.65e26T^{2} \) |
| 61 | \( 1 - 1.56e13T + 6.02e26T^{2} \) |
| 67 | \( 1 - 6.47e13T + 2.46e27T^{2} \) |
| 71 | \( 1 - 1.38e14T + 5.87e27T^{2} \) |
| 73 | \( 1 + 5.76e13T + 8.90e27T^{2} \) |
| 79 | \( 1 - 1.85e14T + 2.91e28T^{2} \) |
| 83 | \( 1 + 3.07e14T + 6.11e28T^{2} \) |
| 89 | \( 1 + 1.81e14T + 1.74e29T^{2} \) |
| 97 | \( 1 + 1.82e14T + 6.33e29T^{2} \) |
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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
−16.77683200436030348164112878375, −16.01719778423616035572737560116, −13.92042686494785765624038293957, −12.43223860759449504643554059912, −11.40073356157614210382640980135, −9.777170891710757936159813134059, −6.51792103893196097256117846195, −5.87007184809743479809345747410, −3.84406424235310651912738150051, −1.05652376614415983558783477407,
1.05652376614415983558783477407, 3.84406424235310651912738150051, 5.87007184809743479809345747410, 6.51792103893196097256117846195, 9.777170891710757936159813134059, 11.40073356157614210382640980135, 12.43223860759449504643554059912, 13.92042686494785765624038293957, 16.01719778423616035572737560116, 16.77683200436030348164112878375
