| L(s) = 1 | − 3-s + 5-s + 3.62·7-s + 9-s − 6.20·11-s − 0.578·13-s − 15-s + 1.42·17-s − 5.62·19-s − 3.62·21-s + 5.62·23-s + 25-s − 27-s + 2·29-s − 2.57·31-s + 6.20·33-s + 3.62·35-s + 7.83·37-s + 0.578·39-s − 5.25·41-s + 7.25·43-s + 45-s + 6.78·47-s + 6.15·49-s − 1.42·51-s + 2·53-s − 6.20·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1.37·7-s + 0.333·9-s − 1.87·11-s − 0.160·13-s − 0.258·15-s + 0.344·17-s − 1.29·19-s − 0.791·21-s + 1.17·23-s + 0.200·25-s − 0.192·27-s + 0.371·29-s − 0.463·31-s + 1.08·33-s + 0.613·35-s + 1.28·37-s + 0.0926·39-s − 0.820·41-s + 1.10·43-s + 0.149·45-s + 0.989·47-s + 0.879·49-s − 0.199·51-s + 0.274·53-s − 0.836·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.719387670\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.719387670\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| good | 7 | \( 1 - 3.62T + 7T^{2} \) |
| 11 | \( 1 + 6.20T + 11T^{2} \) |
| 13 | \( 1 + 0.578T + 13T^{2} \) |
| 17 | \( 1 - 1.42T + 17T^{2} \) |
| 19 | \( 1 + 5.62T + 19T^{2} \) |
| 23 | \( 1 - 5.62T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 2.57T + 31T^{2} \) |
| 37 | \( 1 - 7.83T + 37T^{2} \) |
| 41 | \( 1 + 5.25T + 41T^{2} \) |
| 43 | \( 1 - 7.25T + 43T^{2} \) |
| 47 | \( 1 - 6.78T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 - 2.20T + 59T^{2} \) |
| 61 | \( 1 - 12.4T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 8.41T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 - 5.42T + 79T^{2} \) |
| 83 | \( 1 - 3.25T + 83T^{2} \) |
| 89 | \( 1 - 13.2T + 89T^{2} \) |
| 97 | \( 1 - 4.84T + 97T^{2} \) |
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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
−8.337702114404272222902412693485, −7.79749667720417596638601843407, −7.10357676469451190764930613242, −6.12692934387805100168835692762, −5.27142135551028810300695850719, −5.01402190397445052088750995500, −4.12381235219561182874213259525, −2.69659815563183594262504280023, −2.03621277886311350921908334777, −0.78155673211614980318945705187,
0.78155673211614980318945705187, 2.03621277886311350921908334777, 2.69659815563183594262504280023, 4.12381235219561182874213259525, 5.01402190397445052088750995500, 5.27142135551028810300695850719, 6.12692934387805100168835692762, 7.10357676469451190764930613242, 7.79749667720417596638601843407, 8.337702114404272222902412693485
