| L(s) = 1 | + 0.413·3-s − 0.213·5-s + 2.17·7-s − 2.82·9-s − 0.709·11-s − 0.607·13-s − 0.0885·15-s + 1.74·17-s − 19-s + 0.899·21-s − 9.13·23-s − 4.95·25-s − 2.41·27-s + 1.51·29-s + 2.53·31-s − 0.293·33-s − 0.465·35-s + 6.72·37-s − 0.251·39-s + 4.51·41-s + 3.73·43-s + 0.605·45-s + 5.26·47-s − 2.27·49-s + 0.721·51-s + 11.8·53-s + 0.151·55-s + ⋯ |
| L(s) = 1 | + 0.238·3-s − 0.0956·5-s + 0.821·7-s − 0.942·9-s − 0.213·11-s − 0.168·13-s − 0.0228·15-s + 0.422·17-s − 0.229·19-s + 0.196·21-s − 1.90·23-s − 0.990·25-s − 0.464·27-s + 0.281·29-s + 0.455·31-s − 0.0511·33-s − 0.0786·35-s + 1.10·37-s − 0.0402·39-s + 0.705·41-s + 0.569·43-s + 0.0902·45-s + 0.768·47-s − 0.324·49-s + 0.101·51-s + 1.63·53-s + 0.0204·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.880003225\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.880003225\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + T \) |
| 79 | \( 1 + T \) |
| good | 3 | \( 1 - 0.413T + 3T^{2} \) |
| 5 | \( 1 + 0.213T + 5T^{2} \) |
| 7 | \( 1 - 2.17T + 7T^{2} \) |
| 11 | \( 1 + 0.709T + 11T^{2} \) |
| 13 | \( 1 + 0.607T + 13T^{2} \) |
| 17 | \( 1 - 1.74T + 17T^{2} \) |
| 23 | \( 1 + 9.13T + 23T^{2} \) |
| 29 | \( 1 - 1.51T + 29T^{2} \) |
| 31 | \( 1 - 2.53T + 31T^{2} \) |
| 37 | \( 1 - 6.72T + 37T^{2} \) |
| 41 | \( 1 - 4.51T + 41T^{2} \) |
| 43 | \( 1 - 3.73T + 43T^{2} \) |
| 47 | \( 1 - 5.26T + 47T^{2} \) |
| 53 | \( 1 - 11.8T + 53T^{2} \) |
| 59 | \( 1 - 10.8T + 59T^{2} \) |
| 61 | \( 1 - 3.66T + 61T^{2} \) |
| 67 | \( 1 - 11.1T + 67T^{2} \) |
| 71 | \( 1 - 8.44T + 71T^{2} \) |
| 73 | \( 1 - 13.3T + 73T^{2} \) |
| 83 | \( 1 + 1.71T + 83T^{2} \) |
| 89 | \( 1 + 7.69T + 89T^{2} \) |
| 97 | \( 1 + 12.6T + 97T^{2} \) |
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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
−8.156093958411020217327102442032, −7.64403457620360485761315320648, −6.65019730497724171905277978055, −5.75243917303810820872665833474, −5.41256214278378938848732149431, −4.28570828392064542046056528438, −3.79045007115843568987666174429, −2.57719893712088412829673686838, −2.07844718222358847785600645442, −0.69468460093595083302537426745,
0.69468460093595083302537426745, 2.07844718222358847785600645442, 2.57719893712088412829673686838, 3.79045007115843568987666174429, 4.28570828392064542046056528438, 5.41256214278378938848732149431, 5.75243917303810820872665833474, 6.65019730497724171905277978055, 7.64403457620360485761315320648, 8.156093958411020217327102442032
