| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.069612674\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.069612674\) |
| \(L(\frac{3}{2})\) |
|
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
−16.08290016336319, −15.09899147909658, −14.93881838565298, −14.29606188758307, −14.01581061157693, −13.36464310502616, −12.39062196794915, −12.15884142327556, −11.53544565078424, −11.12853443318762, −10.20504752183517, −9.605202302033285, −9.057018491267000, −8.701716425527106, −7.860032745507402, −7.415626991539509, −6.723608877877547, −6.083643271468407, −5.324312731031795, −4.549157187434392, −3.807107230396754, −3.362188141204069, −2.460517415261194, −1.719631792978963, −0.5964336565036037,
0.5964336565036037, 1.719631792978963, 2.460517415261194, 3.362188141204069, 3.807107230396754, 4.549157187434392, 5.324312731031795, 6.083643271468407, 6.723608877877547, 7.415626991539509, 7.860032745507402, 8.701716425527106, 9.057018491267000, 9.605202302033285, 10.20504752183517, 11.12853443318762, 11.53544565078424, 12.15884142327556, 12.39062196794915, 13.36464310502616, 14.01581061157693, 14.29606188758307, 14.93881838565298, 15.09899147909658, 16.08290016336319
