| L(s) = 1 | + 2.29·3-s + 3.91·5-s − 7-s + 2.25·9-s + 11-s − 13-s + 8.97·15-s + 5.29·17-s + 2.52·19-s − 2.29·21-s − 4.13·23-s + 10.3·25-s − 1.70·27-s + 6.65·29-s − 3.98·31-s + 2.29·33-s − 3.91·35-s + 2.57·37-s − 2.29·39-s + 4.72·41-s − 7.81·43-s + 8.82·45-s + 0.823·47-s + 49-s + 12.1·51-s + 0.474·53-s + 3.91·55-s + ⋯ |
| L(s) = 1 | + 1.32·3-s + 1.75·5-s − 0.377·7-s + 0.751·9-s + 0.301·11-s − 0.277·13-s + 2.31·15-s + 1.28·17-s + 0.579·19-s − 0.500·21-s − 0.861·23-s + 2.06·25-s − 0.328·27-s + 1.23·29-s − 0.715·31-s + 0.399·33-s − 0.661·35-s + 0.423·37-s − 0.367·39-s + 0.737·41-s − 1.19·43-s + 1.31·45-s + 0.120·47-s + 0.142·49-s + 1.69·51-s + 0.0651·53-s + 0.528·55-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
| \(L(1)\) |
\(\approx\) |
\(5.067888719\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.067888719\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.29T + 3T^{2} \) |
| 5 | \( 1 - 3.91T + 5T^{2} \) |
| 17 | \( 1 - 5.29T + 17T^{2} \) |
| 19 | \( 1 - 2.52T + 19T^{2} \) |
| 23 | \( 1 + 4.13T + 23T^{2} \) |
| 29 | \( 1 - 6.65T + 29T^{2} \) |
| 31 | \( 1 + 3.98T + 31T^{2} \) |
| 37 | \( 1 - 2.57T + 37T^{2} \) |
| 41 | \( 1 - 4.72T + 41T^{2} \) |
| 43 | \( 1 + 7.81T + 43T^{2} \) |
| 47 | \( 1 - 0.823T + 47T^{2} \) |
| 53 | \( 1 - 0.474T + 53T^{2} \) |
| 59 | \( 1 - 9.39T + 59T^{2} \) |
| 61 | \( 1 - 6.43T + 61T^{2} \) |
| 67 | \( 1 + 13.5T + 67T^{2} \) |
| 71 | \( 1 - 11.2T + 71T^{2} \) |
| 73 | \( 1 - 10.9T + 73T^{2} \) |
| 79 | \( 1 - 14.4T + 79T^{2} \) |
| 83 | \( 1 + 15.6T + 83T^{2} \) |
| 89 | \( 1 + 16.0T + 89T^{2} \) |
| 97 | \( 1 - 1.43T + 97T^{2} \) |
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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
−8.020249816265219784523060571899, −7.14066421039395044284246894453, −6.48838119178765581944527705013, −5.69670594487850162837065138879, −5.24926478092810691575214138562, −4.09318965785764381139155149101, −3.24845646739175221189469683537, −2.65188517407452467510325753386, −1.96697049031867533096641116206, −1.12030375127877512008095934409,
1.12030375127877512008095934409, 1.96697049031867533096641116206, 2.65188517407452467510325753386, 3.24845646739175221189469683537, 4.09318965785764381139155149101, 5.24926478092810691575214138562, 5.69670594487850162837065138879, 6.48838119178765581944527705013, 7.14066421039395044284246894453, 8.020249816265219784523060571899
