L(s) = 1 | + 2-s − 0.289·3-s + 4-s − 1.50·5-s − 0.289·6-s − 2.57·7-s + 8-s − 2.91·9-s − 1.50·10-s − 5.43·11-s − 0.289·12-s − 1.47·13-s − 2.57·14-s + 0.434·15-s + 16-s + 1.57·17-s − 2.91·18-s − 4.90·19-s − 1.50·20-s + 0.746·21-s − 5.43·22-s + 6.21·23-s − 0.289·24-s − 2.74·25-s − 1.47·26-s + 1.71·27-s − 2.57·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.167·3-s + 0.5·4-s − 0.672·5-s − 0.118·6-s − 0.975·7-s + 0.353·8-s − 0.972·9-s − 0.475·10-s − 1.63·11-s − 0.0835·12-s − 0.409·13-s − 0.689·14-s + 0.112·15-s + 0.250·16-s + 0.382·17-s − 0.687·18-s − 1.12·19-s − 0.336·20-s + 0.162·21-s − 1.15·22-s + 1.29·23-s − 0.0590·24-s − 0.548·25-s − 0.289·26-s + 0.329·27-s − 0.487·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9743984823\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9743984823\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| 97 | \( 1 + T \) |
good | 3 | \( 1 + 0.289T + 3T^{2} \) |
| 5 | \( 1 + 1.50T + 5T^{2} \) |
| 7 | \( 1 + 2.57T + 7T^{2} \) |
| 11 | \( 1 + 5.43T + 11T^{2} \) |
| 13 | \( 1 + 1.47T + 13T^{2} \) |
| 17 | \( 1 - 1.57T + 17T^{2} \) |
| 19 | \( 1 + 4.90T + 19T^{2} \) |
| 23 | \( 1 - 6.21T + 23T^{2} \) |
| 29 | \( 1 - 5.70T + 29T^{2} \) |
| 37 | \( 1 + 5.74T + 37T^{2} \) |
| 41 | \( 1 - 7.81T + 41T^{2} \) |
| 43 | \( 1 + 5.87T + 43T^{2} \) |
| 47 | \( 1 + 5.18T + 47T^{2} \) |
| 53 | \( 1 + 8.60T + 53T^{2} \) |
| 59 | \( 1 - 10.6T + 59T^{2} \) |
| 61 | \( 1 - 12.5T + 61T^{2} \) |
| 67 | \( 1 + 6.89T + 67T^{2} \) |
| 71 | \( 1 - 16.6T + 71T^{2} \) |
| 73 | \( 1 + 9.02T + 73T^{2} \) |
| 79 | \( 1 + 1.95T + 79T^{2} \) |
| 83 | \( 1 - 15.3T + 83T^{2} \) |
| 89 | \( 1 + 15.2T + 89T^{2} \) |
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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
−8.170051627319543397870742871531, −7.19713624590466226974098141629, −6.64207575063580305673414967098, −5.81887260522681820776179600334, −5.20262634534124515580458004604, −4.55915755911250275598044806280, −3.47891790657846624875904722501, −2.97110375675145320822546270501, −2.25881548736054592172506030559, −0.43230650839830601057224978541,
0.43230650839830601057224978541, 2.25881548736054592172506030559, 2.97110375675145320822546270501, 3.47891790657846624875904722501, 4.55915755911250275598044806280, 5.20262634534124515580458004604, 5.81887260522681820776179600334, 6.64207575063580305673414967098, 7.19713624590466226974098141629, 8.170051627319543397870742871531