| L(s) = 1 | − 184.·5-s − 576.·7-s + 729·9-s + 2.64e3·11-s + 3.12e3·17-s − 6.85e3·19-s + 2.06e4·23-s + 1.83e4·25-s + 1.06e5·35-s + 1.32e5·43-s − 1.34e5·45-s − 1.30e5·47-s + 2.15e5·49-s − 4.87e5·55-s + 4.18e5·61-s − 4.20e5·63-s + 3.93e5·73-s − 1.52e6·77-s + 5.31e5·81-s − 1.13e6·83-s − 5.76e5·85-s + 1.26e6·95-s + 1.92e6·99-s + 2.06e6·101-s − 3.80e6·115-s − 1.80e6·119-s + ⋯ |
| L(s) = 1 | − 1.47·5-s − 1.68·7-s + 0.999·9-s + 1.98·11-s + 0.635·17-s − 19-s + 1.69·23-s + 1.17·25-s + 2.48·35-s + 1.66·43-s − 1.47·45-s − 1.25·47-s + 1.82·49-s − 2.93·55-s + 1.84·61-s − 1.68·63-s + 1.01·73-s − 3.34·77-s + 81-s − 1.97·83-s − 0.937·85-s + 1.47·95-s + 1.98·99-s + 1.99·101-s − 2.49·115-s − 1.06·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.277827266\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.277827266\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + 6.85e3T \) |
| good | 3 | \( 1 - 729T^{2} \) |
| 5 | \( 1 + 184.T + 1.56e4T^{2} \) |
| 7 | \( 1 + 576.T + 1.17e5T^{2} \) |
| 11 | \( 1 - 2.64e3T + 1.77e6T^{2} \) |
| 13 | \( 1 - 4.82e6T^{2} \) |
| 17 | \( 1 - 3.12e3T + 2.41e7T^{2} \) |
| 23 | \( 1 - 2.06e4T + 1.48e8T^{2} \) |
| 29 | \( 1 - 5.94e8T^{2} \) |
| 31 | \( 1 - 8.87e8T^{2} \) |
| 37 | \( 1 - 2.56e9T^{2} \) |
| 41 | \( 1 - 4.75e9T^{2} \) |
| 43 | \( 1 - 1.32e5T + 6.32e9T^{2} \) |
| 47 | \( 1 + 1.30e5T + 1.07e10T^{2} \) |
| 53 | \( 1 - 2.21e10T^{2} \) |
| 59 | \( 1 - 4.21e10T^{2} \) |
| 61 | \( 1 - 4.18e5T + 5.15e10T^{2} \) |
| 67 | \( 1 - 9.04e10T^{2} \) |
| 71 | \( 1 - 1.28e11T^{2} \) |
| 73 | \( 1 - 3.93e5T + 1.51e11T^{2} \) |
| 79 | \( 1 - 2.43e11T^{2} \) |
| 83 | \( 1 + 1.13e6T + 3.26e11T^{2} \) |
| 89 | \( 1 - 4.96e11T^{2} \) |
| 97 | \( 1 - 8.32e11T^{2} \) |
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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
−12.92910546163084820562973390869, −12.34481914792019523880085169072, −11.23558566226203095796494135437, −9.812378182070511461324973175288, −8.840090718763794422822185015426, −7.20578489496551891896268427215, −6.51534767134144540042328342593, −4.20333397752856320841372488735, −3.42926797368717717599525481559, −0.818571483744313695778946047792,
0.818571483744313695778946047792, 3.42926797368717717599525481559, 4.20333397752856320841372488735, 6.51534767134144540042328342593, 7.20578489496551891896268427215, 8.840090718763794422822185015426, 9.812378182070511461324973175288, 11.23558566226203095796494135437, 12.34481914792019523880085169072, 12.92910546163084820562973390869
