| L(s) = 1 | + 5-s + 2.38·11-s − 1.38·13-s + 0.831·17-s + 4.21·19-s − 5.59·23-s + 25-s + 10.0·29-s + 31-s + 4.59·37-s − 8.81·41-s − 1.83·43-s + 12.4·47-s − 1.66·53-s + 2.38·55-s − 8.38·59-s − 5.21·61-s − 1.38·65-s + 1.38·67-s + 3.61·71-s + 12.5·73-s − 11.8·79-s + 3.93·83-s + 0.831·85-s + 0.720·89-s + 4.21·95-s + 2.44·97-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.718·11-s − 0.383·13-s + 0.201·17-s + 0.967·19-s − 1.16·23-s + 0.200·25-s + 1.86·29-s + 0.179·31-s + 0.756·37-s − 1.37·41-s − 0.279·43-s + 1.81·47-s − 0.228·53-s + 0.321·55-s − 1.09·59-s − 0.667·61-s − 0.171·65-s + 0.169·67-s + 0.429·71-s + 1.47·73-s − 1.33·79-s + 0.431·83-s + 0.0901·85-s + 0.0763·89-s + 0.432·95-s + 0.248·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.455142372\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.455142372\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 2.38T + 11T^{2} \) |
| 13 | \( 1 + 1.38T + 13T^{2} \) |
| 17 | \( 1 - 0.831T + 17T^{2} \) |
| 19 | \( 1 - 4.21T + 19T^{2} \) |
| 23 | \( 1 + 5.59T + 23T^{2} \) |
| 29 | \( 1 - 10.0T + 29T^{2} \) |
| 31 | \( 1 - T + 31T^{2} \) |
| 37 | \( 1 - 4.59T + 37T^{2} \) |
| 41 | \( 1 + 8.81T + 41T^{2} \) |
| 43 | \( 1 + 1.83T + 43T^{2} \) |
| 47 | \( 1 - 12.4T + 47T^{2} \) |
| 53 | \( 1 + 1.66T + 53T^{2} \) |
| 59 | \( 1 + 8.38T + 59T^{2} \) |
| 61 | \( 1 + 5.21T + 61T^{2} \) |
| 67 | \( 1 - 1.38T + 67T^{2} \) |
| 71 | \( 1 - 3.61T + 71T^{2} \) |
| 73 | \( 1 - 12.5T + 73T^{2} \) |
| 79 | \( 1 + 11.8T + 79T^{2} \) |
| 83 | \( 1 - 3.93T + 83T^{2} \) |
| 89 | \( 1 - 0.720T + 89T^{2} \) |
| 97 | \( 1 - 2.44T + 97T^{2} \) |
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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
−7.76239864099451788843394873897, −7.01637212322477587139592870709, −6.35424212361151576251348381412, −5.78441298302536490108963138472, −4.95298538854508861537967294971, −4.31325650796092204126393789766, −3.41148171930809142463615973774, −2.64879437549679009576422397682, −1.71473796686944068190747985689, −0.78733594341089206319635152057,
0.78733594341089206319635152057, 1.71473796686944068190747985689, 2.64879437549679009576422397682, 3.41148171930809142463615973774, 4.31325650796092204126393789766, 4.95298538854508861537967294971, 5.78441298302536490108963138472, 6.35424212361151576251348381412, 7.01637212322477587139592870709, 7.76239864099451788843394873897
