| L(s) = 1 | − 5-s − 3.43·7-s − 2.63·11-s − 13-s − 7.84·17-s − 0.794·19-s − 3.43·23-s + 25-s + 4.06·29-s − 9.27·31-s + 3.43·35-s − 0.636·37-s + 4.63·41-s − 2.41·43-s + 5.27·47-s + 4.77·49-s − 11.4·53-s + 2.63·55-s − 12.1·59-s − 11.4·61-s + 65-s + 6.41·67-s − 10.6·71-s + 16.0·73-s + 9.04·77-s − 10.6·79-s − 16.1·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.29·7-s − 0.795·11-s − 0.277·13-s − 1.90·17-s − 0.182·19-s − 0.715·23-s + 0.200·25-s + 0.755·29-s − 1.66·31-s + 0.580·35-s − 0.104·37-s + 0.724·41-s − 0.367·43-s + 0.769·47-s + 0.682·49-s − 1.57·53-s + 0.355·55-s − 1.57·59-s − 1.47·61-s + 0.124·65-s + 0.783·67-s − 1.26·71-s + 1.88·73-s + 1.03·77-s − 1.19·79-s − 1.77·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2643557888\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2643557888\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 3.43T + 7T^{2} \) |
| 11 | \( 1 + 2.63T + 11T^{2} \) |
| 17 | \( 1 + 7.84T + 17T^{2} \) |
| 19 | \( 1 + 0.794T + 19T^{2} \) |
| 23 | \( 1 + 3.43T + 23T^{2} \) |
| 29 | \( 1 - 4.06T + 29T^{2} \) |
| 31 | \( 1 + 9.27T + 31T^{2} \) |
| 37 | \( 1 + 0.636T + 37T^{2} \) |
| 41 | \( 1 - 4.63T + 41T^{2} \) |
| 43 | \( 1 + 2.41T + 43T^{2} \) |
| 47 | \( 1 - 5.27T + 47T^{2} \) |
| 53 | \( 1 + 11.4T + 53T^{2} \) |
| 59 | \( 1 + 12.1T + 59T^{2} \) |
| 61 | \( 1 + 11.4T + 61T^{2} \) |
| 67 | \( 1 - 6.41T + 67T^{2} \) |
| 71 | \( 1 + 10.6T + 71T^{2} \) |
| 73 | \( 1 - 16.0T + 73T^{2} \) |
| 79 | \( 1 + 10.6T + 79T^{2} \) |
| 83 | \( 1 + 16.1T + 83T^{2} \) |
| 89 | \( 1 + 8.63T + 89T^{2} \) |
| 97 | \( 1 - 12.2T + 97T^{2} \) |
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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
−7.58173401753043446306720366488, −7.05683133430576395078735229746, −6.33557961625537766257476393799, −5.83918260973735601012768489600, −4.78672836915990166233987461791, −4.25731865645310902294310123214, −3.36392874001707909126387384295, −2.71609486082682473995320311968, −1.87444017477721779166961382651, −0.22876645191488613691994856342,
0.22876645191488613691994856342, 1.87444017477721779166961382651, 2.71609486082682473995320311968, 3.36392874001707909126387384295, 4.25731865645310902294310123214, 4.78672836915990166233987461791, 5.83918260973735601012768489600, 6.33557961625537766257476393799, 7.05683133430576395078735229746, 7.58173401753043446306720366488
