| L(s) = 1 | + 1.39·2-s + 3-s − 0.0513·4-s + 3.05·5-s + 1.39·6-s − 7-s − 2.86·8-s + 9-s + 4.25·10-s + 5.31·11-s − 0.0513·12-s − 3.31·13-s − 1.39·14-s + 3.05·15-s − 3.89·16-s + 0.948·17-s + 1.39·18-s − 19-s − 0.156·20-s − 21-s + 7.41·22-s + 1.31·23-s − 2.86·24-s + 4.31·25-s − 4.62·26-s + 27-s + 0.0513·28-s + ⋯ |
| L(s) = 1 | + 0.987·2-s + 0.577·3-s − 0.0256·4-s + 1.36·5-s + 0.569·6-s − 0.377·7-s − 1.01·8-s + 0.333·9-s + 1.34·10-s + 1.60·11-s − 0.0148·12-s − 0.918·13-s − 0.373·14-s + 0.787·15-s − 0.973·16-s + 0.230·17-s + 0.329·18-s − 0.229·19-s − 0.0350·20-s − 0.218·21-s + 1.58·22-s + 0.273·23-s − 0.584·24-s + 0.862·25-s − 0.906·26-s + 0.192·27-s + 0.00970·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 399 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 399 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.761506607\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.761506607\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 - 1.39T + 2T^{2} \) |
| 5 | \( 1 - 3.05T + 5T^{2} \) |
| 11 | \( 1 - 5.31T + 11T^{2} \) |
| 13 | \( 1 + 3.31T + 13T^{2} \) |
| 17 | \( 1 - 0.948T + 17T^{2} \) |
| 23 | \( 1 - 1.31T + 23T^{2} \) |
| 29 | \( 1 + 7.84T + 29T^{2} \) |
| 31 | \( 1 + 4.79T + 31T^{2} \) |
| 37 | \( 1 + 8.62T + 37T^{2} \) |
| 41 | \( 1 - 11.3T + 41T^{2} \) |
| 43 | \( 1 - 3.20T + 43T^{2} \) |
| 47 | \( 1 + 5.84T + 47T^{2} \) |
| 53 | \( 1 + 2.77T + 53T^{2} \) |
| 59 | \( 1 + 8.20T + 59T^{2} \) |
| 61 | \( 1 - 12.6T + 61T^{2} \) |
| 67 | \( 1 + 4.51T + 67T^{2} \) |
| 71 | \( 1 + 13.5T + 71T^{2} \) |
| 73 | \( 1 + 4.10T + 73T^{2} \) |
| 79 | \( 1 + 8.20T + 79T^{2} \) |
| 83 | \( 1 + 6.36T + 83T^{2} \) |
| 89 | \( 1 - 15.5T + 89T^{2} \) |
| 97 | \( 1 - 11.5T + 97T^{2} \) |
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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
−11.53824443193448835165892721793, −10.13747278087485160967299857856, −9.234744011646566970448844250378, −9.057157262933332291332071696904, −7.26177513710809638175320118686, −6.26386634874263879136623074131, −5.48811314175446666034803490128, −4.28894803860738802833264060397, −3.24014193388489653431117946010, −1.92556801346990797493138724393,
1.92556801346990797493138724393, 3.24014193388489653431117946010, 4.28894803860738802833264060397, 5.48811314175446666034803490128, 6.26386634874263879136623074131, 7.26177513710809638175320118686, 9.057157262933332291332071696904, 9.234744011646566970448844250378, 10.13747278087485160967299857856, 11.53824443193448835165892721793
