| L(s) = 1 | + 3-s − 2.85·5-s + 7-s + 9-s + 3·11-s − 2.61·13-s − 2.85·15-s + 7.47·17-s − 4.23·19-s + 21-s − 23-s + 3.14·25-s + 27-s − 5·29-s + 10.2·31-s + 3·33-s − 2.85·35-s + 0.236·37-s − 2.61·39-s + 3·41-s + 0.618·43-s − 2.85·45-s + 1.23·47-s + 49-s + 7.47·51-s + 13.3·53-s − 8.56·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.27·5-s + 0.377·7-s + 0.333·9-s + 0.904·11-s − 0.726·13-s − 0.736·15-s + 1.81·17-s − 0.971·19-s + 0.218·21-s − 0.208·23-s + 0.629·25-s + 0.192·27-s − 0.928·29-s + 1.83·31-s + 0.522·33-s − 0.482·35-s + 0.0388·37-s − 0.419·39-s + 0.468·41-s + 0.0942·43-s − 0.425·45-s + 0.180·47-s + 0.142·49-s + 1.04·51-s + 1.83·53-s − 1.15·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.842608656\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.842608656\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 + 2.85T + 5T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 + 2.61T + 13T^{2} \) |
| 17 | \( 1 - 7.47T + 17T^{2} \) |
| 19 | \( 1 + 4.23T + 19T^{2} \) |
| 29 | \( 1 + 5T + 29T^{2} \) |
| 31 | \( 1 - 10.2T + 31T^{2} \) |
| 37 | \( 1 - 0.236T + 37T^{2} \) |
| 41 | \( 1 - 3T + 41T^{2} \) |
| 43 | \( 1 - 0.618T + 43T^{2} \) |
| 47 | \( 1 - 1.23T + 47T^{2} \) |
| 53 | \( 1 - 13.3T + 53T^{2} \) |
| 59 | \( 1 - 6.56T + 59T^{2} \) |
| 61 | \( 1 - 1.85T + 61T^{2} \) |
| 67 | \( 1 + 13.3T + 67T^{2} \) |
| 71 | \( 1 - 5.38T + 71T^{2} \) |
| 73 | \( 1 - 11.9T + 73T^{2} \) |
| 79 | \( 1 + 0.527T + 79T^{2} \) |
| 83 | \( 1 - 7.76T + 83T^{2} \) |
| 89 | \( 1 - 13.5T + 89T^{2} \) |
| 97 | \( 1 - 15.9T + 97T^{2} \) |
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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
−9.069867361502579942271053463303, −8.256683035674711739936586861682, −7.75148335482796847821590972041, −7.12230985919452991536831036150, −6.08838358951654932862855142673, −4.93308876280857548302633182269, −4.06746777793584371417086405870, −3.50749200502440451111547487037, −2.32862714825028516788227863840, −0.914667651901002173468624619614,
0.914667651901002173468624619614, 2.32862714825028516788227863840, 3.50749200502440451111547487037, 4.06746777793584371417086405870, 4.93308876280857548302633182269, 6.08838358951654932862855142673, 7.12230985919452991536831036150, 7.75148335482796847821590972041, 8.256683035674711739936586861682, 9.069867361502579942271053463303
