| L(s) = 1 | + 19.2·3-s − 33.5·5-s − 67.1·7-s + 128.·9-s − 121·11-s − 504.·13-s − 646.·15-s + 984.·17-s + 281.·19-s − 1.29e3·21-s − 359.·23-s − 1.99e3·25-s − 2.20e3·27-s + 5.02e3·29-s + 7.01e3·31-s − 2.33e3·33-s + 2.25e3·35-s + 5.24e3·37-s − 9.72e3·39-s − 1.38e4·41-s + 2.01e4·43-s − 4.30e3·45-s − 6.78e3·47-s − 1.22e4·49-s + 1.89e4·51-s + 2.72e4·53-s + 4.05e3·55-s + ⋯ |
| L(s) = 1 | + 1.23·3-s − 0.600·5-s − 0.518·7-s + 0.528·9-s − 0.301·11-s − 0.828·13-s − 0.741·15-s + 0.826·17-s + 0.178·19-s − 0.640·21-s − 0.141·23-s − 0.639·25-s − 0.583·27-s + 1.10·29-s + 1.31·31-s − 0.372·33-s + 0.310·35-s + 0.629·37-s − 1.02·39-s − 1.28·41-s + 1.66·43-s − 0.316·45-s − 0.447·47-s − 0.731·49-s + 1.02·51-s + 1.33·53-s + 0.180·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.487367393\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.487367393\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 + 121T \) |
| good | 3 | \( 1 - 19.2T + 243T^{2} \) |
| 5 | \( 1 + 33.5T + 3.12e3T^{2} \) |
| 7 | \( 1 + 67.1T + 1.68e4T^{2} \) |
| 13 | \( 1 + 504.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 984.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 281.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 359.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 5.02e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 7.01e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 5.24e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.38e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.01e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 6.78e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.72e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.90e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.40e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.32e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.42e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.19e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.63e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.94e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 3.13e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.34e5T + 8.58e9T^{2} \) |
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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
−9.738979556084376896834493882915, −8.681381960347730654940120687157, −7.977907351200612859533277499045, −7.39301827240293199703959053750, −6.25659110578381914397103172377, −4.99905357547837363451960990712, −3.88225025640208308197991571819, −3.05720567650389293819942072751, −2.26750516379119393300338834812, −0.67460626137236580394911963060,
0.67460626137236580394911963060, 2.26750516379119393300338834812, 3.05720567650389293819942072751, 3.88225025640208308197991571819, 4.99905357547837363451960990712, 6.25659110578381914397103172377, 7.39301827240293199703959053750, 7.977907351200612859533277499045, 8.681381960347730654940120687157, 9.738979556084376896834493882915
