| L(s) = 1 | + 238.·5-s + 33.2·7-s + 729·9-s + 1.58e3·11-s + 6.50e3·17-s + 6.85e3·19-s − 2.06e4·23-s + 4.12e4·25-s + 7.91e3·35-s − 1.04e4·43-s + 1.73e5·45-s − 2.05e5·47-s − 1.16e5·49-s + 3.77e5·55-s − 3.61e5·61-s + 2.42e4·63-s − 7.78e5·73-s + 5.25e4·77-s + 5.31e5·81-s + 1.13e6·83-s + 1.55e6·85-s + 1.63e6·95-s + 1.15e6·99-s + 2.06e6·101-s − 4.91e6·115-s + 2.16e5·119-s + ⋯ |
| L(s) = 1 | + 1.90·5-s + 0.0968·7-s + 0.999·9-s + 1.18·11-s + 1.32·17-s + 19-s − 1.69·23-s + 2.63·25-s + 0.184·35-s − 0.131·43-s + 1.90·45-s − 1.97·47-s − 0.990·49-s + 2.26·55-s − 1.59·61-s + 0.0968·63-s − 1.99·73-s + 0.115·77-s + 81-s + 1.97·83-s + 2.52·85-s + 1.90·95-s + 1.18·99-s + 1.99·101-s − 3.23·115-s + 0.128·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(4.099753051\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.099753051\) |
| \(L(4)\) |
|
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 | \( 1 - 238.T + 1.56e4T^{2} \) |
| 7 | \( 1 - 33.2T + 1.17e5T^{2} \) |
| 11 | \( 1 - 1.58e3T + 1.77e6T^{2} \) |
| 13 | \( 1 - 4.82e6T^{2} \) |
| 17 | \( 1 - 6.50e3T + 2.41e7T^{2} \) |
| 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 | \( 1 + 1.04e4T + 6.32e9T^{2} \) |
| 47 | \( 1 + 2.05e5T + 1.07e10T^{2} \) |
| 53 | \( 1 - 2.21e10T^{2} \) |
| 59 | \( 1 - 4.21e10T^{2} \) |
| 61 | \( 1 + 3.61e5T + 5.15e10T^{2} \) |
| 67 | \( 1 - 9.04e10T^{2} \) |
| 71 | \( 1 - 1.28e11T^{2} \) |
| 73 | \( 1 + 7.78e5T + 1.51e11T^{2} \) |
| 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
−10.24570070772479838333031863155, −9.840689802201950905164466409692, −9.156691717379835533965203016662, −7.74581617009666701658670766252, −6.55532881516015027762061194481, −5.87302911062769265082332620743, −4.78964074962069973759460246322, −3.35920799227001125258870102661, −1.80929623559780080515127044967, −1.24484335328824013542240165516,
1.24484335328824013542240165516, 1.80929623559780080515127044967, 3.35920799227001125258870102661, 4.78964074962069973759460246322, 5.87302911062769265082332620743, 6.55532881516015027762061194481, 7.74581617009666701658670766252, 9.156691717379835533965203016662, 9.840689802201950905164466409692, 10.24570070772479838333031863155
