| L(s) = 1 | + 254.·3-s + 151.·5-s − 9.40e3·7-s + 4.48e4·9-s + 5.69e4·11-s − 6.08e4·13-s + 3.85e4·15-s + 8.35e4·17-s − 1.00e6·19-s − 2.39e6·21-s − 1.35e6·23-s − 1.93e6·25-s + 6.39e6·27-s + 3.12e6·29-s − 2.97e6·31-s + 1.44e7·33-s − 1.42e6·35-s + 6.81e5·37-s − 1.54e7·39-s − 4.09e6·41-s − 1.00e7·43-s + 6.79e6·45-s − 2.54e7·47-s + 4.81e7·49-s + 2.12e7·51-s − 3.14e7·53-s + 8.63e6·55-s + ⋯ |
| L(s) = 1 | + 1.81·3-s + 0.108·5-s − 1.48·7-s + 2.27·9-s + 1.17·11-s − 0.591·13-s + 0.196·15-s + 0.242·17-s − 1.77·19-s − 2.68·21-s − 1.01·23-s − 0.988·25-s + 2.31·27-s + 0.820·29-s − 0.578·31-s + 2.12·33-s − 0.160·35-s + 0.0597·37-s − 1.07·39-s − 0.226·41-s − 0.446·43-s + 0.247·45-s − 0.760·47-s + 1.19·49-s + 0.439·51-s − 0.547·53-s + 0.127·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 272 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 272 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 17 | \( 1 - 8.35e4T \) |
| good | 3 | \( 1 - 254.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 151.T + 1.95e6T^{2} \) |
| 7 | \( 1 + 9.40e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 5.69e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 6.08e4T + 1.06e10T^{2} \) |
| 19 | \( 1 + 1.00e6T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.35e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 3.12e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 2.97e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 6.81e5T + 1.29e14T^{2} \) |
| 41 | \( 1 + 4.09e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 1.00e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 2.54e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 3.14e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 9.03e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 9.87e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.32e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 4.18e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 4.80e7T + 5.88e16T^{2} \) |
| 79 | \( 1 + 3.49e7T + 1.19e17T^{2} \) |
| 83 | \( 1 - 2.05e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 2.03e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.24e9T + 7.60e17T^{2} \) |
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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
−9.694331779260664150544832035851, −8.949420085942844988788594194125, −8.126350528873989249176950310449, −6.97411565129365264436370481903, −6.23912457506327493775934867597, −4.22851101844603148962842218568, −3.58698010081133977128265830218, −2.58638826292399751228567292442, −1.68242776538171664461169980204, 0,
1.68242776538171664461169980204, 2.58638826292399751228567292442, 3.58698010081133977128265830218, 4.22851101844603148962842218568, 6.23912457506327493775934867597, 6.97411565129365264436370481903, 8.126350528873989249176950310449, 8.949420085942844988788594194125, 9.694331779260664150544832035851
