| L(s) = 1 | + 0.0181·3-s + 5·5-s − 10.0·7-s − 26.9·9-s − 32.0·11-s + 58.3·13-s + 0.0908·15-s + 49.3·17-s − 70.7·19-s − 0.183·21-s − 23·23-s + 25·25-s − 0.980·27-s − 25.9·29-s − 144.·31-s − 0.582·33-s − 50.4·35-s + 7.35·37-s + 1.05·39-s + 151.·41-s + 245.·43-s − 134.·45-s + 123.·47-s − 241.·49-s + 0.896·51-s − 251.·53-s − 160.·55-s + ⋯ |
| L(s) = 1 | + 0.00349·3-s + 0.447·5-s − 0.544·7-s − 0.999·9-s − 0.878·11-s + 1.24·13-s + 0.00156·15-s + 0.703·17-s − 0.853·19-s − 0.00190·21-s − 0.208·23-s + 0.200·25-s − 0.00699·27-s − 0.165·29-s − 0.838·31-s − 0.00307·33-s − 0.243·35-s + 0.0326·37-s + 0.00434·39-s + 0.578·41-s + 0.872·43-s − 0.447·45-s + 0.382·47-s − 0.703·49-s + 0.00246·51-s − 0.650·53-s − 0.392·55-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
| \(L(2)\) |
\(\approx\) |
\(1.609472179\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.609472179\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 0.0181T + 27T^{2} \) |
| 7 | \( 1 + 10.0T + 343T^{2} \) |
| 11 | \( 1 + 32.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 58.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 49.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 70.7T + 6.85e3T^{2} \) |
| 29 | \( 1 + 25.9T + 2.43e4T^{2} \) |
| 31 | \( 1 + 144.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 7.35T + 5.06e4T^{2} \) |
| 41 | \( 1 - 151.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 245.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 123.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 251.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 271.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 49.9T + 2.26e5T^{2} \) |
| 67 | \( 1 - 602.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 513.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 403.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 284.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 300.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 672.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.07e3T + 9.12e5T^{2} \) |
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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
−8.870498058187985275259362365178, −8.207551189991280364160745918854, −7.38534899224613729221052567461, −6.13705629482101840890972325147, −5.94971122144707716159327318197, −4.95279586857505916758282004981, −3.71878758758768188515943516262, −2.95622444013472081082784267488, −1.95809157833503062367994309952, −0.57518495177276648831978686307,
0.57518495177276648831978686307, 1.95809157833503062367994309952, 2.95622444013472081082784267488, 3.71878758758768188515943516262, 4.95279586857505916758282004981, 5.94971122144707716159327318197, 6.13705629482101840890972325147, 7.38534899224613729221052567461, 8.207551189991280364160745918854, 8.870498058187985275259362365178
