| L(s) = 1 | + 0.299·3-s − 0.576·5-s + 7-s − 2.91·9-s − 5.77·11-s − 0.368·13-s − 0.172·15-s − 1.05·17-s − 19-s + 0.299·21-s − 8.78·23-s − 4.66·25-s − 1.77·27-s + 6.98·29-s − 10.9·31-s − 1.73·33-s − 0.576·35-s + 1.44·37-s − 0.110·39-s + 2.56·41-s + 9.34·43-s + 1.67·45-s + 7.28·47-s + 49-s − 0.317·51-s − 4.88·53-s + 3.32·55-s + ⋯ |
| L(s) = 1 | + 0.173·3-s − 0.257·5-s + 0.377·7-s − 0.970·9-s − 1.73·11-s − 0.102·13-s − 0.0446·15-s − 0.256·17-s − 0.229·19-s + 0.0654·21-s − 1.83·23-s − 0.933·25-s − 0.341·27-s + 1.29·29-s − 1.96·31-s − 0.301·33-s − 0.0974·35-s + 0.237·37-s − 0.0177·39-s + 0.400·41-s + 1.42·43-s + 0.250·45-s + 1.06·47-s + 0.142·49-s − 0.0444·51-s − 0.670·53-s + 0.448·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8776070847\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8776070847\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 0.299T + 3T^{2} \) |
| 5 | \( 1 + 0.576T + 5T^{2} \) |
| 11 | \( 1 + 5.77T + 11T^{2} \) |
| 13 | \( 1 + 0.368T + 13T^{2} \) |
| 17 | \( 1 + 1.05T + 17T^{2} \) |
| 23 | \( 1 + 8.78T + 23T^{2} \) |
| 29 | \( 1 - 6.98T + 29T^{2} \) |
| 31 | \( 1 + 10.9T + 31T^{2} \) |
| 37 | \( 1 - 1.44T + 37T^{2} \) |
| 41 | \( 1 - 2.56T + 41T^{2} \) |
| 43 | \( 1 - 9.34T + 43T^{2} \) |
| 47 | \( 1 - 7.28T + 47T^{2} \) |
| 53 | \( 1 + 4.88T + 53T^{2} \) |
| 59 | \( 1 - 14.0T + 59T^{2} \) |
| 61 | \( 1 + 13.4T + 61T^{2} \) |
| 67 | \( 1 + 4.08T + 67T^{2} \) |
| 71 | \( 1 - 0.491T + 71T^{2} \) |
| 73 | \( 1 - 3.68T + 73T^{2} \) |
| 79 | \( 1 - 10.2T + 79T^{2} \) |
| 83 | \( 1 - 10.5T + 83T^{2} \) |
| 89 | \( 1 + 7.64T + 89T^{2} \) |
| 97 | \( 1 - 14.7T + 97T^{2} \) |
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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
−7.71148764938283935943194795128, −7.49247288761635381603946499723, −6.18655449555459958022086352519, −5.74501174272654137605067057977, −5.06804204390368304979538250892, −4.24458480496782987437763484434, −3.46994800240237662162067129618, −2.47742488531946074387077725823, −2.08686681627952409218540748002, −0.42043788726760691910190159070,
0.42043788726760691910190159070, 2.08686681627952409218540748002, 2.47742488531946074387077725823, 3.46994800240237662162067129618, 4.24458480496782987437763484434, 5.06804204390368304979538250892, 5.74501174272654137605067057977, 6.18655449555459958022086352519, 7.49247288761635381603946499723, 7.71148764938283935943194795128
