| L(s) = 1 | + 3-s + 5-s + 3.64·7-s + 9-s − 5.64·11-s − 5.64·13-s + 15-s − 4·17-s + 19-s + 3.64·21-s + 1.29·23-s + 25-s + 27-s − 6.93·29-s − 6·31-s − 5.64·33-s + 3.64·35-s − 1.64·37-s − 5.64·39-s − 4.35·41-s − 0.354·43-s + 45-s − 9.29·47-s + 6.29·49-s − 4·51-s + 0.708·53-s − 5.64·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1.37·7-s + 0.333·9-s − 1.70·11-s − 1.56·13-s + 0.258·15-s − 0.970·17-s + 0.229·19-s + 0.795·21-s + 0.269·23-s + 0.200·25-s + 0.192·27-s − 1.28·29-s − 1.07·31-s − 0.982·33-s + 0.616·35-s − 0.270·37-s − 0.904·39-s − 0.680·41-s − 0.0540·43-s + 0.149·45-s − 1.35·47-s + 0.898·49-s − 0.560·51-s + 0.0973·53-s − 0.761·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 - 3.64T + 7T^{2} \) |
| 11 | \( 1 + 5.64T + 11T^{2} \) |
| 13 | \( 1 + 5.64T + 13T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 23 | \( 1 - 1.29T + 23T^{2} \) |
| 29 | \( 1 + 6.93T + 29T^{2} \) |
| 31 | \( 1 + 6T + 31T^{2} \) |
| 37 | \( 1 + 1.64T + 37T^{2} \) |
| 41 | \( 1 + 4.35T + 41T^{2} \) |
| 43 | \( 1 + 0.354T + 43T^{2} \) |
| 47 | \( 1 + 9.29T + 47T^{2} \) |
| 53 | \( 1 - 0.708T + 53T^{2} \) |
| 59 | \( 1 + 0.708T + 59T^{2} \) |
| 61 | \( 1 + 0.708T + 61T^{2} \) |
| 67 | \( 1 + 14.5T + 67T^{2} \) |
| 71 | \( 1 - 3.29T + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 - 14.5T + 79T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 + 1.06T + 89T^{2} \) |
| 97 | \( 1 - 12.9T + 97T^{2} \) |
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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
−7.76177688667222254413923350502, −7.59799929049065973024828667310, −6.70058278207679356763305761096, −5.32764120326536005248220903749, −5.15903143566908425991132953718, −4.38349895758822959219982031548, −3.14208681812751923497089405649, −2.25021389140852050145134855535, −1.80967557397962061215883980847, 0,
1.80967557397962061215883980847, 2.25021389140852050145134855535, 3.14208681812751923497089405649, 4.38349895758822959219982031548, 5.15903143566908425991132953718, 5.32764120326536005248220903749, 6.70058278207679356763305761096, 7.59799929049065973024828667310, 7.76177688667222254413923350502
