| L(s) = 1 | + 10.2·3-s + 5·5-s + 33.8·7-s + 77.4·9-s + 21.0·11-s + 32.8·13-s + 51.0·15-s − 28.7·17-s − 67.2·19-s + 346.·21-s − 23·23-s + 25·25-s + 515.·27-s − 199.·29-s − 2.47·31-s + 214.·33-s + 169.·35-s − 173.·37-s + 335.·39-s + 200.·41-s − 198.·43-s + 387.·45-s − 33.4·47-s + 803.·49-s − 293.·51-s + 556.·53-s + 105.·55-s + ⋯ |
| L(s) = 1 | + 1.96·3-s + 0.447·5-s + 1.82·7-s + 2.86·9-s + 0.576·11-s + 0.701·13-s + 0.879·15-s − 0.409·17-s − 0.812·19-s + 3.59·21-s − 0.208·23-s + 0.200·25-s + 3.67·27-s − 1.28·29-s − 0.0143·31-s + 1.13·33-s + 0.817·35-s − 0.771·37-s + 1.37·39-s + 0.763·41-s − 0.704·43-s + 1.28·45-s − 0.103·47-s + 2.34·49-s − 0.806·51-s + 1.44·53-s + 0.257·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(7.406897967\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.406897967\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| 23 | \( 1 + 23T \) |
| good | 3 | \( 1 - 10.2T + 27T^{2} \) |
| 7 | \( 1 - 33.8T + 343T^{2} \) |
| 11 | \( 1 - 21.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 32.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 28.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 67.2T + 6.85e3T^{2} \) |
| 29 | \( 1 + 199.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 2.47T + 2.97e4T^{2} \) |
| 37 | \( 1 + 173.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 200.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 198.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 33.4T + 1.03e5T^{2} \) |
| 53 | \( 1 - 556.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 234.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 26.2T + 2.26e5T^{2} \) |
| 67 | \( 1 + 190.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 745.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 742.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 715.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 766.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 683.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.17e3T + 9.12e5T^{2} \) |
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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
−8.688986438694964532857457116266, −8.372775729907737059427026583539, −7.54696558381632082315214552157, −6.85946343584779667725133395892, −5.57314245554772193525656760322, −4.37298369289658383907652502022, −4.00053476849062388606329600381, −2.76763016875216247281196840499, −1.79685773660640765526976776014, −1.45372833836443366349240705649,
1.45372833836443366349240705649, 1.79685773660640765526976776014, 2.76763016875216247281196840499, 4.00053476849062388606329600381, 4.37298369289658383907652502022, 5.57314245554772193525656760322, 6.85946343584779667725133395892, 7.54696558381632082315214552157, 8.372775729907737059427026583539, 8.688986438694964532857457116266
