| L(s) = 1 | − 20.9·2-s + 81·3-s − 71.1·4-s + 1.46e3·5-s − 1.70e3·6-s + 1.22e4·8-s + 6.56e3·9-s − 3.07e4·10-s − 3.49e3·11-s − 5.76e3·12-s − 3.96e4·13-s + 1.18e5·15-s − 2.20e5·16-s − 2.05e5·17-s − 1.37e5·18-s + 2.16e5·19-s − 1.04e5·20-s + 7.33e4·22-s − 2.12e6·23-s + 9.91e5·24-s + 1.91e5·25-s + 8.32e5·26-s + 5.31e5·27-s − 1.51e6·29-s − 2.49e6·30-s + 3.64e6·31-s − 1.63e6·32-s + ⋯ |
| L(s) = 1 | − 0.927·2-s + 0.577·3-s − 0.139·4-s + 1.04·5-s − 0.535·6-s + 1.05·8-s + 0.333·9-s − 0.972·10-s − 0.0719·11-s − 0.0802·12-s − 0.385·13-s + 0.605·15-s − 0.841·16-s − 0.595·17-s − 0.309·18-s + 0.380·19-s − 0.145·20-s + 0.0667·22-s − 1.58·23-s + 0.610·24-s + 0.0982·25-s + 0.357·26-s + 0.192·27-s − 0.397·29-s − 0.561·30-s + 0.709·31-s − 0.275·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 81T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 20.9T + 512T^{2} \) |
| 5 | \( 1 - 1.46e3T + 1.95e6T^{2} \) |
| 11 | \( 1 + 3.49e3T + 2.35e9T^{2} \) |
| 13 | \( 1 + 3.96e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 2.05e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 2.16e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 2.12e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 1.51e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 3.64e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 7.13e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 2.60e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 3.54e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 3.71e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 2.33e6T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.36e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 2.11e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.29e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 4.72e7T + 4.58e16T^{2} \) |
| 73 | \( 1 + 2.63e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 3.85e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 5.60e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 1.50e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.06e8T + 7.60e17T^{2} \) |
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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
−10.14050181876089052717816937313, −9.891918890967685087307327598088, −8.828627465739920676876829313859, −8.008216121859671671352555562108, −6.81630403225314800704442486967, −5.41718244128961722190191319511, −4.09535422704396007436238728974, −2.38946847731581775870835617183, −1.46875458028025330270706654149, 0,
1.46875458028025330270706654149, 2.38946847731581775870835617183, 4.09535422704396007436238728974, 5.41718244128961722190191319511, 6.81630403225314800704442486967, 8.008216121859671671352555562108, 8.828627465739920676876829313859, 9.891918890967685087307327598088, 10.14050181876089052717816937313
