| L(s) = 1 | − 24.7·3-s − 36.1·5-s + 370.·9-s − 155.·11-s + 1.15e3·13-s + 894.·15-s + 1.23e3·17-s + 280.·19-s + 3.48e3·23-s − 1.82e3·25-s − 3.16e3·27-s − 5.65e3·29-s + 2.31e3·31-s + 3.85e3·33-s − 2.33e3·37-s − 2.87e4·39-s + 3.81e3·41-s − 3.92e3·43-s − 1.33e4·45-s + 1.11e4·47-s − 3.06e4·51-s − 1.11e4·53-s + 5.61e3·55-s − 6.94e3·57-s + 6.01e3·59-s − 1.48e4·61-s − 4.18e4·65-s + ⋯ |
| L(s) = 1 | − 1.58·3-s − 0.645·5-s + 1.52·9-s − 0.387·11-s + 1.90·13-s + 1.02·15-s + 1.03·17-s + 0.178·19-s + 1.37·23-s − 0.582·25-s − 0.836·27-s − 1.24·29-s + 0.432·31-s + 0.615·33-s − 0.280·37-s − 3.02·39-s + 0.354·41-s − 0.323·43-s − 0.985·45-s + 0.734·47-s − 1.65·51-s − 0.547·53-s + 0.250·55-s − 0.283·57-s + 0.224·59-s − 0.510·61-s − 1.22·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.062019434\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.062019434\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 24.7T + 243T^{2} \) |
| 5 | \( 1 + 36.1T + 3.12e3T^{2} \) |
| 11 | \( 1 + 155.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 1.15e3T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.23e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 280.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.48e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 5.65e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.31e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 2.33e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 3.81e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 3.92e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.11e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.11e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 6.01e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.48e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.29e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.99e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.55e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.08e5T + 3.07e9T^{2} \) |
| 83 | \( 1 + 5.58e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 9.55e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.50e4T + 8.58e9T^{2} \) |
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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
−9.754675289933099969435148336545, −8.615516529979394591816954257210, −7.68109187105090477975704010482, −6.80142494995207337014746130510, −5.86179399695778639830996383989, −5.34238143913230963027094223147, −4.19858221929350980340795769890, −3.30286559667231064156920527134, −1.38770434514426292727618943849, −0.57568601080488398875323522001,
0.57568601080488398875323522001, 1.38770434514426292727618943849, 3.30286559667231064156920527134, 4.19858221929350980340795769890, 5.34238143913230963027094223147, 5.86179399695778639830996383989, 6.80142494995207337014746130510, 7.68109187105090477975704010482, 8.615516529979394591816954257210, 9.754675289933099969435148336545
