L(s) = 1 | − 5-s + 3.37·7-s − 4·11-s + 4.74·13-s − 5.37·17-s + 4·19-s − 23-s + 25-s + 5.37·29-s + 3.37·31-s − 3.37·35-s − 5.37·37-s + 8.11·41-s − 4·43-s + 6.74·47-s + 4.37·49-s − 1.37·53-s + 4·55-s − 8.62·59-s − 2·61-s − 4.74·65-s + 15.3·67-s − 3.37·71-s − 12.7·73-s − 13.4·77-s − 13.4·79-s + 12.8·83-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 1.27·7-s − 1.20·11-s + 1.31·13-s − 1.30·17-s + 0.917·19-s − 0.208·23-s + 0.200·25-s + 0.997·29-s + 0.605·31-s − 0.570·35-s − 0.883·37-s + 1.26·41-s − 0.609·43-s + 0.983·47-s + 0.624·49-s − 0.188·53-s + 0.539·55-s − 1.12·59-s − 0.256·61-s − 0.588·65-s + 1.87·67-s − 0.400·71-s − 1.49·73-s − 1.53·77-s − 1.51·79-s + 1.41·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.111563980\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.111563980\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 7 | \( 1 - 3.37T + 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 - 4.74T + 13T^{2} \) |
| 17 | \( 1 + 5.37T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 29 | \( 1 - 5.37T + 29T^{2} \) |
| 31 | \( 1 - 3.37T + 31T^{2} \) |
| 37 | \( 1 + 5.37T + 37T^{2} \) |
| 41 | \( 1 - 8.11T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 - 6.74T + 47T^{2} \) |
| 53 | \( 1 + 1.37T + 53T^{2} \) |
| 59 | \( 1 + 8.62T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 15.3T + 67T^{2} \) |
| 71 | \( 1 + 3.37T + 71T^{2} \) |
| 73 | \( 1 + 12.7T + 73T^{2} \) |
| 79 | \( 1 + 13.4T + 79T^{2} \) |
| 83 | \( 1 - 12.8T + 83T^{2} \) |
| 89 | \( 1 - 4.74T + 89T^{2} \) |
| 97 | \( 1 - 15.4T + 97T^{2} \) |
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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
−7.79922287046638019129428262320, −7.33805938904478591516676571468, −6.39305431292893731603544899779, −5.69536524381127841629776588179, −4.84616735157180718890589725613, −4.47476889922948518254120293156, −3.50699099925323460065889132064, −2.64064791865709465665266864340, −1.73972496766131965581097556829, −0.73272593811362229314989823617,
0.73272593811362229314989823617, 1.73972496766131965581097556829, 2.64064791865709465665266864340, 3.50699099925323460065889132064, 4.47476889922948518254120293156, 4.84616735157180718890589725613, 5.69536524381127841629776588179, 6.39305431292893731603544899779, 7.33805938904478591516676571468, 7.79922287046638019129428262320