| L(s) = 1 | − 5.31·3-s − 9.41·5-s − 16.2·7-s + 1.29·9-s + 8.71·11-s + 43.4·13-s + 50.0·15-s − 124.·17-s − 19·19-s + 86.4·21-s − 71.6·23-s − 36.3·25-s + 136.·27-s − 279.·29-s − 252.·31-s − 46.3·33-s + 152.·35-s − 17.9·37-s − 230.·39-s − 466.·41-s − 340.·43-s − 12.2·45-s + 305.·47-s − 78.9·49-s + 661.·51-s + 293.·53-s − 82.0·55-s + ⋯ |
| L(s) = 1 | − 1.02·3-s − 0.841·5-s − 0.877·7-s + 0.0480·9-s + 0.238·11-s + 0.926·13-s + 0.861·15-s − 1.77·17-s − 0.229·19-s + 0.898·21-s − 0.649·23-s − 0.291·25-s + 0.974·27-s − 1.79·29-s − 1.46·31-s − 0.244·33-s + 0.738·35-s − 0.0797·37-s − 0.948·39-s − 1.77·41-s − 1.20·43-s − 0.0404·45-s + 0.947·47-s − 0.230·49-s + 1.81·51-s + 0.760·53-s − 0.201·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.06783152151\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06783152151\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + 19T \) |
| good | 3 | \( 1 + 5.31T + 27T^{2} \) |
| 5 | \( 1 + 9.41T + 125T^{2} \) |
| 7 | \( 1 + 16.2T + 343T^{2} \) |
| 11 | \( 1 - 8.71T + 1.33e3T^{2} \) |
| 13 | \( 1 - 43.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 124.T + 4.91e3T^{2} \) |
| 23 | \( 1 + 71.6T + 1.21e4T^{2} \) |
| 29 | \( 1 + 279.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 252.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 17.9T + 5.06e4T^{2} \) |
| 41 | \( 1 + 466.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 340.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 305.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 293.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 408.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 334.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 754.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 758.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 393.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 91.7T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.39e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 931.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 282.T + 9.12e5T^{2} \) |
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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
−9.236787258366545570523927682213, −8.675945444699099502989086718773, −7.58772325153700171430648223099, −6.64520218408357266938905961185, −6.15443253329361969807660029169, −5.20424591357216618156214789626, −4.07453337581117737292220530870, −3.43737350868508072064526359157, −1.86633473527730582244975384912, −0.13392894877217713884090639556,
0.13392894877217713884090639556, 1.86633473527730582244975384912, 3.43737350868508072064526359157, 4.07453337581117737292220530870, 5.20424591357216618156214789626, 6.15443253329361969807660029169, 6.64520218408357266938905961185, 7.58772325153700171430648223099, 8.675945444699099502989086718773, 9.236787258366545570523927682213
