| L(s) = 1 | + 2.88·3-s + 0.398·5-s − 3.33·7-s + 5.30·9-s − 3.34·11-s + 4.80·13-s + 1.14·15-s + 1.16·17-s − 5.31·19-s − 9.60·21-s − 2.25·23-s − 4.84·25-s + 6.63·27-s − 2.97·29-s − 1.24·31-s − 9.62·33-s − 1.32·35-s − 3.41·37-s + 13.8·39-s − 2.23·41-s − 0.269·43-s + 2.11·45-s − 2.36·47-s + 4.12·49-s + 3.34·51-s + 1.57·53-s − 1.33·55-s + ⋯ |
| L(s) = 1 | + 1.66·3-s + 0.178·5-s − 1.26·7-s + 1.76·9-s − 1.00·11-s + 1.33·13-s + 0.296·15-s + 0.281·17-s − 1.21·19-s − 2.09·21-s − 0.470·23-s − 0.968·25-s + 1.27·27-s − 0.553·29-s − 0.223·31-s − 1.67·33-s − 0.224·35-s − 0.562·37-s + 2.21·39-s − 0.349·41-s − 0.0410·43-s + 0.314·45-s − 0.345·47-s + 0.588·49-s + 0.468·51-s + 0.216·53-s − 0.179·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 503 | \( 1 - T \) |
| good | 3 | \( 1 - 2.88T + 3T^{2} \) |
| 5 | \( 1 - 0.398T + 5T^{2} \) |
| 7 | \( 1 + 3.33T + 7T^{2} \) |
| 11 | \( 1 + 3.34T + 11T^{2} \) |
| 13 | \( 1 - 4.80T + 13T^{2} \) |
| 17 | \( 1 - 1.16T + 17T^{2} \) |
| 19 | \( 1 + 5.31T + 19T^{2} \) |
| 23 | \( 1 + 2.25T + 23T^{2} \) |
| 29 | \( 1 + 2.97T + 29T^{2} \) |
| 31 | \( 1 + 1.24T + 31T^{2} \) |
| 37 | \( 1 + 3.41T + 37T^{2} \) |
| 41 | \( 1 + 2.23T + 41T^{2} \) |
| 43 | \( 1 + 0.269T + 43T^{2} \) |
| 47 | \( 1 + 2.36T + 47T^{2} \) |
| 53 | \( 1 - 1.57T + 53T^{2} \) |
| 59 | \( 1 + 2.60T + 59T^{2} \) |
| 61 | \( 1 - 2.08T + 61T^{2} \) |
| 67 | \( 1 + 6.92T + 67T^{2} \) |
| 71 | \( 1 - 5.98T + 71T^{2} \) |
| 73 | \( 1 + 6.47T + 73T^{2} \) |
| 79 | \( 1 + 7.35T + 79T^{2} \) |
| 83 | \( 1 - 3.33T + 83T^{2} \) |
| 89 | \( 1 + 5.26T + 89T^{2} \) |
| 97 | \( 1 - 10.9T + 97T^{2} \) |
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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
−7.71915683240679684899678500512, −6.87645273607746063715242821167, −6.19881125962268545967277724620, −5.53052886640916898538418128895, −4.28294170822666194496328099619, −3.66433356666860633401094719763, −3.14923919859636929928572520639, −2.36608050472041195557329047019, −1.62751564435166567735806897781, 0,
1.62751564435166567735806897781, 2.36608050472041195557329047019, 3.14923919859636929928572520639, 3.66433356666860633401094719763, 4.28294170822666194496328099619, 5.53052886640916898538418128895, 6.19881125962268545967277724620, 6.87645273607746063715242821167, 7.71915683240679684899678500512
