| L(s) = 1 | − 3.16i·3-s + 2.23·5-s − 4.24i·7-s − 7.00·9-s − 7.07i·15-s − 13.4·21-s − 1.41i·23-s + 5.00·25-s + 12.6i·27-s − 8.94·29-s − 9.48i·35-s + 12·41-s + 3.16i·43-s − 15.6·45-s − 9.89i·47-s + ⋯ |
| L(s) = 1 | − 1.82i·3-s + 0.999·5-s − 1.60i·7-s − 2.33·9-s − 1.82i·15-s − 2.92·21-s − 0.294i·23-s + 1.00·25-s + 2.43i·27-s − 1.66·29-s − 1.60i·35-s + 1.87·41-s + 0.482i·43-s − 2.33·45-s − 1.44i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.662165901\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.662165901\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 2.23T \) |
| good | 3 | \( 1 + 3.16iT - 3T^{2} \) |
| 7 | \( 1 + 4.24iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 1.41iT - 23T^{2} \) |
| 29 | \( 1 + 8.94T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 - 12T + 41T^{2} \) |
| 43 | \( 1 - 3.16iT - 43T^{2} \) |
| 47 | \( 1 + 9.89iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 13.4T + 61T^{2} \) |
| 67 | \( 1 - 15.8iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 9.48iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−9.173011658134479185233357235828, −8.211579459080945494931493065755, −7.37180054615200398591522069490, −6.96529908559581923470473206272, −6.15379024255503077825596239705, −5.35570244715184207686346759183, −3.92763752514989788616977354505, −2.60175771206566757867170950348, −1.62362903636296284698723335247, −0.68658930390005499480121379532,
2.18544267703346915664342748151, 3.01490255900182227104995763061, 4.11638414089062825535332761058, 5.24452195390248714795373551901, 5.55366799139059837885331082955, 6.33202653307794440330219547185, 7.962920476369171613868324707218, 9.026621347890859949703816920751, 9.258677909627388581685072499676, 9.816230869561932725957320111121
