| L(s) = 1 | + 17.4·3-s − 79.4·5-s − 132.·7-s + 62.5·9-s − 311.·11-s + 901.·13-s − 1.38e3·15-s − 157.·17-s − 361·19-s − 2.32e3·21-s + 2.52e3·23-s + 3.18e3·25-s − 3.15e3·27-s + 4.73e3·29-s + 6.58e3·31-s − 5.44e3·33-s + 1.05e4·35-s + 8.50e3·37-s + 1.57e4·39-s + 1.97e4·41-s − 1.09e4·43-s − 4.96e3·45-s − 1.50e4·47-s + 860.·49-s − 2.75e3·51-s + 2.16e4·53-s + 2.47e4·55-s + ⋯ |
| L(s) = 1 | + 1.12·3-s − 1.42·5-s − 1.02·7-s + 0.257·9-s − 0.776·11-s + 1.47·13-s − 1.59·15-s − 0.132·17-s − 0.229·19-s − 1.14·21-s + 0.994·23-s + 1.01·25-s − 0.832·27-s + 1.04·29-s + 1.23·31-s − 0.870·33-s + 1.45·35-s + 1.02·37-s + 1.65·39-s + 1.83·41-s − 0.905·43-s − 0.365·45-s − 0.996·47-s + 0.0512·49-s − 0.148·51-s + 1.06·53-s + 1.10·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.802567424\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.802567424\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + 361T \) |
| good | 3 | \( 1 - 17.4T + 243T^{2} \) |
| 5 | \( 1 + 79.4T + 3.12e3T^{2} \) |
| 7 | \( 1 + 132.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 311.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 901.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 157.T + 1.41e6T^{2} \) |
| 23 | \( 1 - 2.52e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 4.73e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 6.58e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 8.50e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.97e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.09e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.50e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.16e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.06e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 6.15e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 6.27e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.53e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.85e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.10e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 4.10e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.70e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.39e5T + 8.58e9T^{2} \) |
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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
−10.93165832182223764812876442185, −9.818005606219827942198950453815, −8.640800240319336635556969107455, −8.259212872510445370234943734948, −7.24321853065369824588815479157, −6.11499031609867749539925671049, −4.39732647365177943184950124265, −3.41567886088548281575416680871, −2.75935346345354813806250921147, −0.70026264921606149208752509872,
0.70026264921606149208752509872, 2.75935346345354813806250921147, 3.41567886088548281575416680871, 4.39732647365177943184950124265, 6.11499031609867749539925671049, 7.24321853065369824588815479157, 8.259212872510445370234943734948, 8.640800240319336635556969107455, 9.818005606219827942198950453815, 10.93165832182223764812876442185
