| L(s) = 1 | − 81·3-s − 2.68e3·5-s + 950.·7-s + 6.56e3·9-s + 8.53e4·11-s + 1.24e5·13-s + 2.17e5·15-s − 2.14e4·17-s − 2.43e5·19-s − 7.69e4·21-s − 7.06e5·23-s + 5.26e6·25-s − 5.31e5·27-s + 3.90e6·29-s + 8.04e6·31-s − 6.91e6·33-s − 2.55e6·35-s + 1.67e7·37-s − 1.01e7·39-s − 3.31e7·41-s − 3.30e7·43-s − 1.76e7·45-s − 3.29e7·47-s − 3.94e7·49-s + 1.73e6·51-s + 5.61e7·53-s − 2.29e8·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.92·5-s + 0.149·7-s + 0.333·9-s + 1.75·11-s + 1.21·13-s + 1.11·15-s − 0.0622·17-s − 0.427·19-s − 0.0863·21-s − 0.526·23-s + 2.69·25-s − 0.192·27-s + 1.02·29-s + 1.56·31-s − 1.01·33-s − 0.287·35-s + 1.46·37-s − 0.700·39-s − 1.83·41-s − 1.47·43-s − 0.640·45-s − 0.985·47-s − 0.977·49-s + 0.0359·51-s + 0.978·53-s − 3.38·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.406453776\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.406453776\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 81T \) |
| good | 5 | \( 1 + 2.68e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 950.T + 4.03e7T^{2} \) |
| 11 | \( 1 - 8.53e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.24e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + 2.14e4T + 1.18e11T^{2} \) |
| 19 | \( 1 + 2.43e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 7.06e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 3.90e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 8.04e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.67e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 3.31e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 3.30e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 3.29e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 5.61e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.09e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 3.77e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 9.13e7T + 2.72e16T^{2} \) |
| 71 | \( 1 - 2.44e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.30e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 4.26e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 2.32e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 2.89e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 2.74e8T + 7.60e17T^{2} \) |
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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
−9.924312732836687591780970952292, −8.503551698951009983112372220231, −8.219612763490884652252698955420, −6.80630557224876863623270387374, −6.39287817832682821440961234313, −4.71995788765678205806170904210, −4.05052466920498709014475812946, −3.31049197798413088209502059114, −1.36510112690037064318345365199, −0.58201593581893872336733478882,
0.58201593581893872336733478882, 1.36510112690037064318345365199, 3.31049197798413088209502059114, 4.05052466920498709014475812946, 4.71995788765678205806170904210, 6.39287817832682821440961234313, 6.80630557224876863623270387374, 8.219612763490884652252698955420, 8.503551698951009983112372220231, 9.924312732836687591780970952292
