| L(s) = 1 | + 0.674·3-s − 4.42·5-s − 7-s − 2.54·9-s + 11-s − 13-s − 2.98·15-s + 4.81·17-s − 5.29·19-s − 0.674·21-s + 1.44·23-s + 14.5·25-s − 3.74·27-s + 7.93·29-s + 8.10·31-s + 0.674·33-s + 4.42·35-s − 6.99·37-s − 0.674·39-s + 7.05·41-s − 7.92·43-s + 11.2·45-s + 1.44·47-s + 49-s + 3.25·51-s − 8.55·53-s − 4.42·55-s + ⋯ |
| L(s) = 1 | + 0.389·3-s − 1.97·5-s − 0.377·7-s − 0.848·9-s + 0.301·11-s − 0.277·13-s − 0.770·15-s + 1.16·17-s − 1.21·19-s − 0.147·21-s + 0.302·23-s + 2.91·25-s − 0.720·27-s + 1.47·29-s + 1.45·31-s + 0.117·33-s + 0.747·35-s − 1.15·37-s − 0.108·39-s + 1.10·41-s − 1.20·43-s + 1.67·45-s + 0.211·47-s + 0.142·49-s + 0.455·51-s − 1.17·53-s − 0.596·55-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 | \( 1 - 0.674T + 3T^{2} \) |
| 5 | \( 1 + 4.42T + 5T^{2} \) |
| 17 | \( 1 - 4.81T + 17T^{2} \) |
| 19 | \( 1 + 5.29T + 19T^{2} \) |
| 23 | \( 1 - 1.44T + 23T^{2} \) |
| 29 | \( 1 - 7.93T + 29T^{2} \) |
| 31 | \( 1 - 8.10T + 31T^{2} \) |
| 37 | \( 1 + 6.99T + 37T^{2} \) |
| 41 | \( 1 - 7.05T + 41T^{2} \) |
| 43 | \( 1 + 7.92T + 43T^{2} \) |
| 47 | \( 1 - 1.44T + 47T^{2} \) |
| 53 | \( 1 + 8.55T + 53T^{2} \) |
| 59 | \( 1 + 0.0353T + 59T^{2} \) |
| 61 | \( 1 - 9.90T + 61T^{2} \) |
| 67 | \( 1 + 1.89T + 67T^{2} \) |
| 71 | \( 1 - 9.06T + 71T^{2} \) |
| 73 | \( 1 + 12.8T + 73T^{2} \) |
| 79 | \( 1 - 11.6T + 79T^{2} \) |
| 83 | \( 1 - 9.53T + 83T^{2} \) |
| 89 | \( 1 + 3.76T + 89T^{2} \) |
| 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
−7.71477936474994147544028262148, −6.81457097074801395362958321016, −6.36877331505546045059457065131, −5.20664283068982938670405706711, −4.53103258612177207826459550413, −3.78054676226467346678582352814, −3.20039773924253852443454490890, −2.57886791464118657542958795947, −0.982913487393907003824736734847, 0,
0.982913487393907003824736734847, 2.57886791464118657542958795947, 3.20039773924253852443454490890, 3.78054676226467346678582352814, 4.53103258612177207826459550413, 5.20664283068982938670405706711, 6.36877331505546045059457065131, 6.81457097074801395362958321016, 7.71477936474994147544028262148
