| L(s) = 1 | − 3-s + 2.78·5-s − 2.60·7-s + 9-s − 2.13·11-s − 1.66·13-s − 2.78·15-s + 0.462·17-s + 1.45·19-s + 2.60·21-s + 1.82·23-s + 2.75·25-s − 27-s − 2.76·29-s + 1.47·31-s + 2.13·33-s − 7.25·35-s + 11.2·37-s + 1.66·39-s + 4.20·41-s − 10.5·43-s + 2.78·45-s − 9.69·47-s − 0.219·49-s − 0.462·51-s + 1.64·53-s − 5.94·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.24·5-s − 0.984·7-s + 0.333·9-s − 0.643·11-s − 0.461·13-s − 0.719·15-s + 0.112·17-s + 0.333·19-s + 0.568·21-s + 0.380·23-s + 0.551·25-s − 0.192·27-s − 0.514·29-s + 0.265·31-s + 0.371·33-s − 1.22·35-s + 1.84·37-s + 0.266·39-s + 0.656·41-s − 1.60·43-s + 0.415·45-s − 1.41·47-s − 0.0313·49-s − 0.0648·51-s + 0.226·53-s − 0.801·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7248 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 151 | \( 1 + T \) |
| good | 5 | \( 1 - 2.78T + 5T^{2} \) |
| 7 | \( 1 + 2.60T + 7T^{2} \) |
| 11 | \( 1 + 2.13T + 11T^{2} \) |
| 13 | \( 1 + 1.66T + 13T^{2} \) |
| 17 | \( 1 - 0.462T + 17T^{2} \) |
| 19 | \( 1 - 1.45T + 19T^{2} \) |
| 23 | \( 1 - 1.82T + 23T^{2} \) |
| 29 | \( 1 + 2.76T + 29T^{2} \) |
| 31 | \( 1 - 1.47T + 31T^{2} \) |
| 37 | \( 1 - 11.2T + 37T^{2} \) |
| 41 | \( 1 - 4.20T + 41T^{2} \) |
| 43 | \( 1 + 10.5T + 43T^{2} \) |
| 47 | \( 1 + 9.69T + 47T^{2} \) |
| 53 | \( 1 - 1.64T + 53T^{2} \) |
| 59 | \( 1 - 11.6T + 59T^{2} \) |
| 61 | \( 1 + 2.46T + 61T^{2} \) |
| 67 | \( 1 + 3.36T + 67T^{2} \) |
| 71 | \( 1 + 0.392T + 71T^{2} \) |
| 73 | \( 1 + 1.20T + 73T^{2} \) |
| 79 | \( 1 - 13.1T + 79T^{2} \) |
| 83 | \( 1 - 7.14T + 83T^{2} \) |
| 89 | \( 1 - 3.77T + 89T^{2} \) |
| 97 | \( 1 + 5.41T + 97T^{2} \) |
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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
−7.48861490075059262627259270784, −6.50923648452780371728137464153, −6.35055938196361837908027960288, −5.37978875979891059039148343663, −5.09000839908142484820348533855, −3.97203460879273286108326288830, −2.96809650609634377838117047344, −2.32691087552749649164943394841, −1.24768826332100172807252513504, 0,
1.24768826332100172807252513504, 2.32691087552749649164943394841, 2.96809650609634377838117047344, 3.97203460879273286108326288830, 5.09000839908142484820348533855, 5.37978875979891059039148343663, 6.35055938196361837908027960288, 6.50923648452780371728137464153, 7.48861490075059262627259270784
