| L(s) = 1 | − 1.70·3-s − 5·5-s − 20.8·7-s − 24.0·9-s − 23.6·11-s + 53.7·13-s + 8.54·15-s + 52.1·17-s − 4.88·19-s + 35.6·21-s + 23·23-s + 25·25-s + 87.3·27-s − 144.·29-s + 288.·31-s + 40.4·33-s + 104.·35-s + 294.·37-s − 91.8·39-s + 333.·41-s + 0.162·43-s + 120.·45-s − 529.·47-s + 92.5·49-s − 89.1·51-s + 116.·53-s + 118.·55-s + ⋯ |
| L(s) = 1 | − 0.329·3-s − 0.447·5-s − 1.12·7-s − 0.891·9-s − 0.648·11-s + 1.14·13-s + 0.147·15-s + 0.743·17-s − 0.0589·19-s + 0.370·21-s + 0.208·23-s + 0.200·25-s + 0.622·27-s − 0.922·29-s + 1.66·31-s + 0.213·33-s + 0.503·35-s + 1.30·37-s − 0.377·39-s + 1.26·41-s + 0.000575·43-s + 0.398·45-s − 1.64·47-s + 0.269·49-s − 0.244·51-s + 0.301·53-s + 0.290·55-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
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| 23 | \( 1 - 23T \) |
| good | 3 | \( 1 + 1.70T + 27T^{2} \) |
| 7 | \( 1 + 20.8T + 343T^{2} \) |
| 11 | \( 1 + 23.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 53.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 52.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 4.88T + 6.85e3T^{2} \) |
| 29 | \( 1 + 144.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 288.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 294.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 333.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 0.162T + 7.95e4T^{2} \) |
| 47 | \( 1 + 529.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 116.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 284.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 120.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 194.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 201.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 913.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 901.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 172.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.57e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.72e3T + 9.12e5T^{2} \) |
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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
−8.372763349176818570488448639867, −7.87082405448029559817726548060, −6.74539620887919318146909497693, −6.07369366339762562569938329337, −5.45750363805994803435041432235, −4.28826105936899744059773592296, −3.30916125212103059318572507276, −2.70015069101442011699805302054, −0.993800067126839626635724396185, 0,
0.993800067126839626635724396185, 2.70015069101442011699805302054, 3.30916125212103059318572507276, 4.28826105936899744059773592296, 5.45750363805994803435041432235, 6.07369366339762562569938329337, 6.74539620887919318146909497693, 7.87082405448029559817726548060, 8.372763349176818570488448639867
