| L(s) = 1 | − 4.65·3-s + 5·5-s + 7·7-s − 5.31·9-s − 52.2·11-s − 30.6·13-s − 23.2·15-s + 37.2·17-s + 80.2·19-s − 32.5·21-s − 25.8·23-s + 25·25-s + 150.·27-s − 20.9·29-s + 314.·31-s + 243.·33-s + 35·35-s − 197.·37-s + 142.·39-s + 11.3·41-s − 33.8·43-s − 26.5·45-s + 361.·47-s + 49·49-s − 173.·51-s − 153.·53-s − 261.·55-s + ⋯ |
| L(s) = 1 | − 0.896·3-s + 0.447·5-s + 0.377·7-s − 0.196·9-s − 1.43·11-s − 0.654·13-s − 0.400·15-s + 0.531·17-s + 0.968·19-s − 0.338·21-s − 0.234·23-s + 0.200·25-s + 1.07·27-s − 0.134·29-s + 1.82·31-s + 1.28·33-s + 0.169·35-s − 0.875·37-s + 0.586·39-s + 0.0432·41-s − 0.119·43-s − 0.0880·45-s + 1.12·47-s + 0.142·49-s − 0.475·51-s − 0.396·53-s − 0.640·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 5T \) |
| 7 | \( 1 - 7T \) |
| good | 3 | \( 1 + 4.65T + 27T^{2} \) |
| 11 | \( 1 + 52.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 30.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 37.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 80.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 25.8T + 1.21e4T^{2} \) |
| 29 | \( 1 + 20.9T + 2.43e4T^{2} \) |
| 31 | \( 1 - 314.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 197.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 11.3T + 6.89e4T^{2} \) |
| 43 | \( 1 + 33.8T + 7.95e4T^{2} \) |
| 47 | \( 1 - 361.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 153.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 616T + 2.05e5T^{2} \) |
| 61 | \( 1 + 15.2T + 2.26e5T^{2} \) |
| 67 | \( 1 + 166.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 952T + 3.57e5T^{2} \) |
| 73 | \( 1 + 148.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 857.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 660.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 45.7T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.68e3T + 9.12e5T^{2} \) |
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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
−8.125286261140618081257650107513, −7.59512288980893705168916775926, −6.61941344608889513758108178889, −5.73631326788966652286486854484, −5.22933164781465375010330711293, −4.63354435332125459815721739072, −3.14001595784909242193341411675, −2.37364060778673079378996164899, −1.05175938807326084276434690972, 0,
1.05175938807326084276434690972, 2.37364060778673079378996164899, 3.14001595784909242193341411675, 4.63354435332125459815721739072, 5.22933164781465375010330711293, 5.73631326788966652286486854484, 6.61941344608889513758108178889, 7.59512288980893705168916775926, 8.125286261140618081257650107513
