| L(s) = 1 | + 2.41·3-s + 0.0537·5-s + 0.349·7-s + 2.82·9-s + 1.73·11-s − 5.19·13-s + 0.129·15-s − 17-s + 6.66·19-s + 0.842·21-s − 8.08·23-s − 4.99·25-s − 0.410·27-s − 4.27·29-s − 9.11·31-s + 4.18·33-s + 0.0187·35-s − 1.81·37-s − 12.5·39-s − 5.44·41-s + 4.67·43-s + 0.152·45-s + 6.77·47-s − 6.87·49-s − 2.41·51-s − 4.55·53-s + 0.0932·55-s + ⋯ |
| L(s) = 1 | + 1.39·3-s + 0.0240·5-s + 0.131·7-s + 0.943·9-s + 0.522·11-s − 1.44·13-s + 0.0335·15-s − 0.242·17-s + 1.52·19-s + 0.183·21-s − 1.68·23-s − 0.999·25-s − 0.0790·27-s − 0.793·29-s − 1.63·31-s + 0.729·33-s + 0.00317·35-s − 0.298·37-s − 2.00·39-s − 0.849·41-s + 0.713·43-s + 0.0226·45-s + 0.988·47-s − 0.982·49-s − 0.338·51-s − 0.625·53-s + 0.0125·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| good | 3 | \( 1 - 2.41T + 3T^{2} \) |
| 5 | \( 1 - 0.0537T + 5T^{2} \) |
| 7 | \( 1 - 0.349T + 7T^{2} \) |
| 11 | \( 1 - 1.73T + 11T^{2} \) |
| 13 | \( 1 + 5.19T + 13T^{2} \) |
| 19 | \( 1 - 6.66T + 19T^{2} \) |
| 23 | \( 1 + 8.08T + 23T^{2} \) |
| 29 | \( 1 + 4.27T + 29T^{2} \) |
| 31 | \( 1 + 9.11T + 31T^{2} \) |
| 37 | \( 1 + 1.81T + 37T^{2} \) |
| 41 | \( 1 + 5.44T + 41T^{2} \) |
| 43 | \( 1 - 4.67T + 43T^{2} \) |
| 47 | \( 1 - 6.77T + 47T^{2} \) |
| 53 | \( 1 + 4.55T + 53T^{2} \) |
| 61 | \( 1 + 5.51T + 61T^{2} \) |
| 67 | \( 1 + 1.57T + 67T^{2} \) |
| 71 | \( 1 + 9.26T + 71T^{2} \) |
| 73 | \( 1 + 12.8T + 73T^{2} \) |
| 79 | \( 1 - 1.61T + 79T^{2} \) |
| 83 | \( 1 - 13.9T + 83T^{2} \) |
| 89 | \( 1 - 4.80T + 89T^{2} \) |
| 97 | \( 1 - 17.1T + 97T^{2} \) |
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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
−7.59634793243311102787604225495, −7.20729870708620775198136148774, −6.06794148559578422798513228644, −5.38949908850479085319214132706, −4.47533898185261295985444960609, −3.71417779059271340552936017534, −3.15851644909013170466717975952, −2.16107208357625987761917703075, −1.71526279348593872247722260416, 0,
1.71526279348593872247722260416, 2.16107208357625987761917703075, 3.15851644909013170466717975952, 3.71417779059271340552936017534, 4.47533898185261295985444960609, 5.38949908850479085319214132706, 6.06794148559578422798513228644, 7.20729870708620775198136148774, 7.59634793243311102787604225495
