L(s) = 1 | + 2-s + 4-s + 5-s + 3·7-s + 8-s + 10-s + 11-s + 3·14-s + 16-s + 5·19-s + 20-s + 22-s + 4·23-s + 25-s + 3·28-s − 10·31-s + 32-s + 3·35-s + 37-s + 5·38-s + 40-s + 6·41-s − 2·43-s + 44-s + 4·46-s − 9·47-s + 2·49-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 1.13·7-s + 0.353·8-s + 0.316·10-s + 0.301·11-s + 0.801·14-s + 1/4·16-s + 1.14·19-s + 0.223·20-s + 0.213·22-s + 0.834·23-s + 1/5·25-s + 0.566·28-s − 1.79·31-s + 0.176·32-s + 0.507·35-s + 0.164·37-s + 0.811·38-s + 0.158·40-s + 0.937·41-s − 0.304·43-s + 0.150·44-s + 0.589·46-s − 1.31·47-s + 2/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.270630214\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.270630214\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 13 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−16.15624674257666, −15.17115706453305, −14.88928742667320, −14.30105374819120, −13.96646395983571, −13.23170801369056, −12.83189181584529, −12.08497429099318, −11.54256242309838, −11.06382732749788, −10.62992894545137, −9.682785195732533, −9.298490092241251, −8.467142695680847, −7.882101089271830, −7.179428509343803, −6.738026006344460, −5.764150369247340, −5.308915199241668, −4.853565299372022, −3.990203174962936, −3.352337255502545, −2.439755181311195, −1.717867667751560, −0.9549144894500126,
0.9549144894500126, 1.717867667751560, 2.439755181311195, 3.352337255502545, 3.990203174962936, 4.853565299372022, 5.308915199241668, 5.764150369247340, 6.738026006344460, 7.179428509343803, 7.882101089271830, 8.467142695680847, 9.298490092241251, 9.682785195732533, 10.62992894545137, 11.06382732749788, 11.54256242309838, 12.08497429099318, 12.83189181584529, 13.23170801369056, 13.96646395983571, 14.30105374819120, 14.88928742667320, 15.17115706453305, 16.15624674257666