| L(s) = 1 | + 1.07·3-s + 2.25·5-s + 7-s − 1.84·9-s − 11-s + 13-s + 2.42·15-s − 3.96·17-s − 1.18·19-s + 1.07·21-s − 1.10·23-s + 0.0634·25-s − 5.20·27-s − 9.51·29-s + 2.11·31-s − 1.07·33-s + 2.25·35-s + 5.93·37-s + 1.07·39-s + 1.04·41-s − 8.80·43-s − 4.14·45-s − 5.75·47-s + 49-s − 4.26·51-s − 8.39·53-s − 2.25·55-s + ⋯ |
| L(s) = 1 | + 0.621·3-s + 1.00·5-s + 0.377·7-s − 0.614·9-s − 0.301·11-s + 0.277·13-s + 0.624·15-s − 0.962·17-s − 0.271·19-s + 0.234·21-s − 0.230·23-s + 0.0126·25-s − 1.00·27-s − 1.76·29-s + 0.379·31-s − 0.187·33-s + 0.380·35-s + 0.975·37-s + 0.172·39-s + 0.162·41-s − 1.34·43-s − 0.618·45-s − 0.839·47-s + 0.142·49-s − 0.597·51-s − 1.15·53-s − 0.303·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 - 1.07T + 3T^{2} \) |
| 5 | \( 1 - 2.25T + 5T^{2} \) |
| 17 | \( 1 + 3.96T + 17T^{2} \) |
| 19 | \( 1 + 1.18T + 19T^{2} \) |
| 23 | \( 1 + 1.10T + 23T^{2} \) |
| 29 | \( 1 + 9.51T + 29T^{2} \) |
| 31 | \( 1 - 2.11T + 31T^{2} \) |
| 37 | \( 1 - 5.93T + 37T^{2} \) |
| 41 | \( 1 - 1.04T + 41T^{2} \) |
| 43 | \( 1 + 8.80T + 43T^{2} \) |
| 47 | \( 1 + 5.75T + 47T^{2} \) |
| 53 | \( 1 + 8.39T + 53T^{2} \) |
| 59 | \( 1 - 1.00T + 59T^{2} \) |
| 61 | \( 1 - 12.2T + 61T^{2} \) |
| 67 | \( 1 + 3.27T + 67T^{2} \) |
| 71 | \( 1 + 9.47T + 71T^{2} \) |
| 73 | \( 1 + 5.44T + 73T^{2} \) |
| 79 | \( 1 - 3.83T + 79T^{2} \) |
| 83 | \( 1 + 2.42T + 83T^{2} \) |
| 89 | \( 1 + 11.4T + 89T^{2} \) |
| 97 | \( 1 + 18.2T + 97T^{2} \) |
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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
−7.62098711345343007043183510202, −6.75195853278749504561647531692, −6.00473480998951588781776522221, −5.53786377868419953623306447123, −4.68782837234700279715218325522, −3.83873339704944005079889928607, −2.94444024766549851719443742089, −2.18921205632082420170389749296, −1.61974678380304888650545112189, 0,
1.61974678380304888650545112189, 2.18921205632082420170389749296, 2.94444024766549851719443742089, 3.83873339704944005079889928607, 4.68782837234700279715218325522, 5.53786377868419953623306447123, 6.00473480998951588781776522221, 6.75195853278749504561647531692, 7.62098711345343007043183510202
