| L(s) = 1 | + 2·3-s + 5-s − 7-s + 9-s − 11-s + 4·13-s + 2·15-s − 17-s − 8·19-s − 2·21-s − 6·23-s + 25-s − 4·27-s − 9·29-s + 2·31-s − 2·33-s − 35-s + 4·37-s + 8·39-s + 9·41-s − 8·43-s + 45-s − 9·47-s − 6·49-s − 2·51-s − 3·53-s − 55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.301·11-s + 1.10·13-s + 0.516·15-s − 0.242·17-s − 1.83·19-s − 0.436·21-s − 1.25·23-s + 1/5·25-s − 0.769·27-s − 1.67·29-s + 0.359·31-s − 0.348·33-s − 0.169·35-s + 0.657·37-s + 1.28·39-s + 1.40·41-s − 1.21·43-s + 0.149·45-s − 1.31·47-s − 6/7·49-s − 0.280·51-s − 0.412·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.547539317\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.547539317\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 18 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−14.42866695289877, −13.74633569863062, −13.26497025169697, −13.01699750400267, −12.62465625958730, −11.72236146081371, −11.03013608303220, −10.93591267940847, −9.914222618770817, −9.744355092359269, −9.167863895777535, −8.475818901444891, −8.235768663903313, −7.852073062850599, −6.864881427815780, −6.491132117336977, −5.886831564332946, −5.414665050940652, −4.411914757286241, −3.925218062022453, −3.456169370146795, −2.716645522708637, −2.045081643764361, −1.760784051170867, −0.4668978485684968,
0.4668978485684968, 1.760784051170867, 2.045081643764361, 2.716645522708637, 3.456169370146795, 3.925218062022453, 4.411914757286241, 5.414665050940652, 5.886831564332946, 6.491132117336977, 6.864881427815780, 7.852073062850599, 8.235768663903313, 8.475818901444891, 9.167863895777535, 9.744355092359269, 9.914222618770817, 10.93591267940847, 11.03013608303220, 11.72236146081371, 12.62465625958730, 13.01699750400267, 13.26497025169697, 13.74633569863062, 14.42866695289877
