| L(s) = 1 | + 1.30·3-s + 5-s + 7-s − 1.30·9-s − 4.60·11-s − 13-s + 1.30·15-s + 6.30·17-s − 6.30·19-s + 1.30·21-s − 8·23-s + 25-s − 5.60·27-s + 1.90·29-s − 0.697·31-s − 6·33-s + 35-s − 3.90·37-s − 1.30·39-s − 4.30·41-s − 2·43-s − 1.30·45-s − 4.60·47-s + 49-s + 8.21·51-s − 2.60·53-s − 4.60·55-s + ⋯ |
| L(s) = 1 | + 0.752·3-s + 0.447·5-s + 0.377·7-s − 0.434·9-s − 1.38·11-s − 0.277·13-s + 0.336·15-s + 1.52·17-s − 1.44·19-s + 0.284·21-s − 1.66·23-s + 0.200·25-s − 1.07·27-s + 0.354·29-s − 0.125·31-s − 1.04·33-s + 0.169·35-s − 0.642·37-s − 0.208·39-s − 0.671·41-s − 0.304·43-s − 0.194·45-s − 0.671·47-s + 0.142·49-s + 1.14·51-s − 0.357·53-s − 0.621·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - 1.30T + 3T^{2} \) |
| 11 | \( 1 + 4.60T + 11T^{2} \) |
| 17 | \( 1 - 6.30T + 17T^{2} \) |
| 19 | \( 1 + 6.30T + 19T^{2} \) |
| 23 | \( 1 + 8T + 23T^{2} \) |
| 29 | \( 1 - 1.90T + 29T^{2} \) |
| 31 | \( 1 + 0.697T + 31T^{2} \) |
| 37 | \( 1 + 3.90T + 37T^{2} \) |
| 41 | \( 1 + 4.30T + 41T^{2} \) |
| 43 | \( 1 + 2T + 43T^{2} \) |
| 47 | \( 1 + 4.60T + 47T^{2} \) |
| 53 | \( 1 + 2.60T + 53T^{2} \) |
| 59 | \( 1 - 11.1T + 59T^{2} \) |
| 61 | \( 1 + 1.21T + 61T^{2} \) |
| 67 | \( 1 - 8.90T + 67T^{2} \) |
| 71 | \( 1 + 2.60T + 71T^{2} \) |
| 73 | \( 1 + 8.60T + 73T^{2} \) |
| 79 | \( 1 - 3.51T + 79T^{2} \) |
| 83 | \( 1 + 11.8T + 83T^{2} \) |
| 89 | \( 1 + 7.30T + 89T^{2} \) |
| 97 | \( 1 + 13.2T + 97T^{2} \) |
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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
−8.202291771525817068894475017954, −7.75544647959697324341432424762, −6.73912472484116101836529748356, −5.71966482278085944371779299034, −5.32770573608866530406381175695, −4.25637542501674371939580414143, −3.28428620990325479946114933008, −2.48758221826284606180735536989, −1.77580286047578491443949644123, 0,
1.77580286047578491443949644123, 2.48758221826284606180735536989, 3.28428620990325479946114933008, 4.25637542501674371939580414143, 5.32770573608866530406381175695, 5.71966482278085944371779299034, 6.73912472484116101836529748356, 7.75544647959697324341432424762, 8.202291771525817068894475017954
