| L(s) = 1 | − 0.684·3-s + 3.65·5-s − 2.53·9-s + 2.37·11-s − 5.46·13-s − 2.49·15-s + 4.86·17-s − 19-s − 1.77·23-s + 8.32·25-s + 3.78·27-s − 6.04·29-s − 1.69·31-s − 1.62·33-s + 5.62·37-s + 3.74·39-s + 2·41-s + 3.10·43-s − 9.24·45-s + 5.23·47-s − 3.32·51-s − 6.46·53-s + 8.67·55-s + 0.684·57-s + 7.52·59-s + 7.75·61-s − 19.9·65-s + ⋯ |
| L(s) = 1 | − 0.395·3-s + 1.63·5-s − 0.843·9-s + 0.716·11-s − 1.51·13-s − 0.644·15-s + 1.17·17-s − 0.229·19-s − 0.370·23-s + 1.66·25-s + 0.728·27-s − 1.12·29-s − 0.305·31-s − 0.283·33-s + 0.924·37-s + 0.599·39-s + 0.312·41-s + 0.474·43-s − 1.37·45-s + 0.762·47-s − 0.465·51-s − 0.888·53-s + 1.17·55-s + 0.0906·57-s + 0.979·59-s + 0.993·61-s − 2.47·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.192927035\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.192927035\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 0.684T + 3T^{2} \) |
| 5 | \( 1 - 3.65T + 5T^{2} \) |
| 11 | \( 1 - 2.37T + 11T^{2} \) |
| 13 | \( 1 + 5.46T + 13T^{2} \) |
| 17 | \( 1 - 4.86T + 17T^{2} \) |
| 23 | \( 1 + 1.77T + 23T^{2} \) |
| 29 | \( 1 + 6.04T + 29T^{2} \) |
| 31 | \( 1 + 1.69T + 31T^{2} \) |
| 37 | \( 1 - 5.62T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 3.10T + 43T^{2} \) |
| 47 | \( 1 - 5.23T + 47T^{2} \) |
| 53 | \( 1 + 6.46T + 53T^{2} \) |
| 59 | \( 1 - 7.52T + 59T^{2} \) |
| 61 | \( 1 - 7.75T + 61T^{2} \) |
| 67 | \( 1 + 2.62T + 67T^{2} \) |
| 71 | \( 1 - 11.1T + 71T^{2} \) |
| 73 | \( 1 + 14.5T + 73T^{2} \) |
| 79 | \( 1 - 0.961T + 79T^{2} \) |
| 83 | \( 1 - 15.0T + 83T^{2} \) |
| 89 | \( 1 + 12.0T + 89T^{2} \) |
| 97 | \( 1 - 6.26T + 97T^{2} \) |
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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
−7.78329535474374901204553337822, −7.12990148358781831220348597714, −6.27535601201304203891048504907, −5.77160471297157202576953310718, −5.33819715330564766219132944171, −4.54720056127998011959163889297, −3.41507709526222035151075243625, −2.50036986497584848428373178316, −1.91699894144696212685307326716, −0.75223176940564700588634755394,
0.75223176940564700588634755394, 1.91699894144696212685307326716, 2.50036986497584848428373178316, 3.41507709526222035151075243625, 4.54720056127998011959163889297, 5.33819715330564766219132944171, 5.77160471297157202576953310718, 6.27535601201304203891048504907, 7.12990148358781831220348597714, 7.78329535474374901204553337822
