| L(s) = 1 | − 3-s − 5-s − 2·9-s + 6·11-s − 4·13-s + 15-s − 2·19-s + 3·23-s + 25-s + 5·27-s − 3·29-s − 8·31-s − 6·33-s − 4·37-s + 4·39-s + 9·41-s + 7·43-s + 2·45-s − 6·53-s − 6·55-s + 2·57-s + 6·59-s + 5·61-s + 4·65-s − 5·67-s − 3·69-s + 6·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 2/3·9-s + 1.80·11-s − 1.10·13-s + 0.258·15-s − 0.458·19-s + 0.625·23-s + 1/5·25-s + 0.962·27-s − 0.557·29-s − 1.43·31-s − 1.04·33-s − 0.657·37-s + 0.640·39-s + 1.40·41-s + 1.06·43-s + 0.298·45-s − 0.824·53-s − 0.809·55-s + 0.264·57-s + 0.781·59-s + 0.640·61-s + 0.496·65-s − 0.610·67-s − 0.361·69-s + 0.712·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.106254507\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.106254507\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−8.697733740758382740553611689848, −7.50284072082334084258384096942, −7.06135577446493813160311680830, −6.21080322340578476338876304348, −5.57381930747296822266094050342, −4.65687374438402552678393155173, −3.95953058177454500668475210908, −3.05422267129199327077040238038, −1.89700733491127948127375430876, −0.61766121148948183247449959512,
0.61766121148948183247449959512, 1.89700733491127948127375430876, 3.05422267129199327077040238038, 3.95953058177454500668475210908, 4.65687374438402552678393155173, 5.57381930747296822266094050342, 6.21080322340578476338876304348, 7.06135577446493813160311680830, 7.50284072082334084258384096942, 8.697733740758382740553611689848
