| L(s) = 1 | + 2.53·3-s + 3.41·9-s − 4.22·11-s + 5.98·13-s − 7.63·17-s + 6.94·19-s − 1.71·23-s + 1.04·27-s + 5.18·29-s − 0.509·31-s − 10.7·33-s + 3.98·37-s + 15.1·39-s + 3.71·41-s + 4.17·43-s + 1.89·47-s − 19.3·51-s + 6.39·53-s + 17.5·57-s − 5.59·59-s − 4.31·61-s + 13.1·67-s − 4.34·69-s + 5.08·71-s − 5.38·73-s − 1.34·79-s − 7.59·81-s + ⋯ |
| L(s) = 1 | + 1.46·3-s + 1.13·9-s − 1.27·11-s + 1.66·13-s − 1.85·17-s + 1.59·19-s − 0.357·23-s + 0.200·27-s + 0.962·29-s − 0.0915·31-s − 1.86·33-s + 0.655·37-s + 2.42·39-s + 0.580·41-s + 0.635·43-s + 0.276·47-s − 2.70·51-s + 0.877·53-s + 2.32·57-s − 0.728·59-s − 0.552·61-s + 1.61·67-s − 0.523·69-s + 0.603·71-s − 0.630·73-s − 0.151·79-s − 0.844·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.696560808\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.696560808\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 2.53T + 3T^{2} \) |
| 11 | \( 1 + 4.22T + 11T^{2} \) |
| 13 | \( 1 - 5.98T + 13T^{2} \) |
| 17 | \( 1 + 7.63T + 17T^{2} \) |
| 19 | \( 1 - 6.94T + 19T^{2} \) |
| 23 | \( 1 + 1.71T + 23T^{2} \) |
| 29 | \( 1 - 5.18T + 29T^{2} \) |
| 31 | \( 1 + 0.509T + 31T^{2} \) |
| 37 | \( 1 - 3.98T + 37T^{2} \) |
| 41 | \( 1 - 3.71T + 41T^{2} \) |
| 43 | \( 1 - 4.17T + 43T^{2} \) |
| 47 | \( 1 - 1.89T + 47T^{2} \) |
| 53 | \( 1 - 6.39T + 53T^{2} \) |
| 59 | \( 1 + 5.59T + 59T^{2} \) |
| 61 | \( 1 + 4.31T + 61T^{2} \) |
| 67 | \( 1 - 13.1T + 67T^{2} \) |
| 71 | \( 1 - 5.08T + 71T^{2} \) |
| 73 | \( 1 + 5.38T + 73T^{2} \) |
| 79 | \( 1 + 1.34T + 79T^{2} \) |
| 83 | \( 1 - 8.70T + 83T^{2} \) |
| 89 | \( 1 - 10.8T + 89T^{2} \) |
| 97 | \( 1 + 13.6T + 97T^{2} \) |
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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
−7.82652985684203956784404326014, −7.22895786296914781658384091223, −6.36593317032140093219071374424, −5.67145292156832161900205219351, −4.74385088340120640313128875920, −4.01513969921099836860815088631, −3.29793224270338873185346638814, −2.64292094840534728812399859398, −2.00313945969853743329480082789, −0.849420885407169467373512811395,
0.849420885407169467373512811395, 2.00313945969853743329480082789, 2.64292094840534728812399859398, 3.29793224270338873185346638814, 4.01513969921099836860815088631, 4.74385088340120640313128875920, 5.67145292156832161900205219351, 6.36593317032140093219071374424, 7.22895786296914781658384091223, 7.82652985684203956784404326014
