| L(s) = 1 | − 3-s + 5-s − 4·7-s + 9-s + 4·11-s + 6·13-s − 15-s + 6·17-s + 4·19-s + 4·21-s + 4·23-s + 25-s − 27-s + 29-s + 8·31-s − 4·33-s − 4·35-s + 2·37-s − 6·39-s − 6·41-s − 4·43-s + 45-s + 9·49-s − 6·51-s − 10·53-s + 4·55-s − 4·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 1.51·7-s + 1/3·9-s + 1.20·11-s + 1.66·13-s − 0.258·15-s + 1.45·17-s + 0.917·19-s + 0.872·21-s + 0.834·23-s + 1/5·25-s − 0.192·27-s + 0.185·29-s + 1.43·31-s − 0.696·33-s − 0.676·35-s + 0.328·37-s − 0.960·39-s − 0.937·41-s − 0.609·43-s + 0.149·45-s + 9/7·49-s − 0.840·51-s − 1.37·53-s + 0.539·55-s − 0.529·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.084485682\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.084485682\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−7.935546860872204670711428817664, −6.90839540942168338017301750484, −6.48760359515946948587938609225, −5.99138804575505018649692721328, −5.35510201506707152774333901791, −4.30119253897216674926579519758, −3.35323349122509343329647461799, −3.12652137631697507885239570266, −1.44004979456005039847621366595, −0.866063470872250409002330873291,
0.866063470872250409002330873291, 1.44004979456005039847621366595, 3.12652137631697507885239570266, 3.35323349122509343329647461799, 4.30119253897216674926579519758, 5.35510201506707152774333901791, 5.99138804575505018649692721328, 6.48760359515946948587938609225, 6.90839540942168338017301750484, 7.935546860872204670711428817664
