| L(s) = 1 | + 4.47·3-s − 5·5-s − 31.3·7-s − 6.99·9-s − 8.94·11-s − 62·13-s − 22.3·15-s − 46·17-s + 107.·19-s − 140·21-s + 192.·23-s + 25·25-s − 152.·27-s − 90·29-s − 152.·31-s − 40.0·33-s + 156.·35-s − 214·37-s − 277.·39-s − 10·41-s − 67.0·43-s + 34.9·45-s + 398.·47-s + 637.·49-s − 205.·51-s − 678·53-s + 44.7·55-s + ⋯ |
| L(s) = 1 | + 0.860·3-s − 0.447·5-s − 1.69·7-s − 0.259·9-s − 0.245·11-s − 1.32·13-s − 0.384·15-s − 0.656·17-s + 1.29·19-s − 1.45·21-s + 1.74·23-s + 0.200·25-s − 1.08·27-s − 0.576·29-s − 0.880·31-s − 0.211·33-s + 0.755·35-s − 0.950·37-s − 1.13·39-s − 0.0380·41-s − 0.237·43-s + 0.115·45-s + 1.23·47-s + 1.85·49-s − 0.564·51-s − 1.75·53-s + 0.109·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| good | 3 | \( 1 - 4.47T + 27T^{2} \) |
| 7 | \( 1 + 31.3T + 343T^{2} \) |
| 11 | \( 1 + 8.94T + 1.33e3T^{2} \) |
| 13 | \( 1 + 62T + 2.19e3T^{2} \) |
| 17 | \( 1 + 46T + 4.91e3T^{2} \) |
| 19 | \( 1 - 107.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 192.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 90T + 2.43e4T^{2} \) |
| 31 | \( 1 + 152.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 214T + 5.06e4T^{2} \) |
| 41 | \( 1 + 10T + 6.89e4T^{2} \) |
| 43 | \( 1 + 67.0T + 7.95e4T^{2} \) |
| 47 | \( 1 - 398.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 678T + 1.48e5T^{2} \) |
| 59 | \( 1 + 411.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 250T + 2.26e5T^{2} \) |
| 67 | \( 1 - 49.1T + 3.00e5T^{2} \) |
| 71 | \( 1 + 366.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 522T + 3.89e5T^{2} \) |
| 79 | \( 1 - 876.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 380.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 970T + 7.04e5T^{2} \) |
| 97 | \( 1 + 934T + 9.12e5T^{2} \) |
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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
−12.14417407189950667783235657394, −10.85092715792755929576333917388, −9.506101788907985375513364111873, −9.120904275493744894244518951720, −7.64245355927736891419684440360, −6.82808572228664515094071631921, −5.25780113204589902706074526859, −3.50053540682691292918108792216, −2.71009757244035338446943905309, 0,
2.71009757244035338446943905309, 3.50053540682691292918108792216, 5.25780113204589902706074526859, 6.82808572228664515094071631921, 7.64245355927736891419684440360, 9.120904275493744894244518951720, 9.506101788907985375513364111873, 10.85092715792755929576333917388, 12.14417407189950667783235657394
