| L(s) = 1 | + 1.30·2-s − 6.30·4-s − 5·5-s − 18.6·8-s − 6.51·10-s + 18.1·11-s + 19.1·13-s + 26.1·16-s − 58.3·17-s + 13.9·19-s + 31.5·20-s + 23.6·22-s − 127.·23-s + 25·25-s + 24.9·26-s + 92.7·29-s + 181.·31-s + 183.·32-s − 75.9·34-s + 145.·37-s + 18.1·38-s + 93.1·40-s + 132.·41-s − 200.·43-s − 114.·44-s − 165.·46-s − 23.2·47-s + ⋯ |
| L(s) = 1 | + 0.460·2-s − 0.787·4-s − 0.447·5-s − 0.823·8-s − 0.205·10-s + 0.498·11-s + 0.408·13-s + 0.408·16-s − 0.832·17-s + 0.168·19-s + 0.352·20-s + 0.229·22-s − 1.15·23-s + 0.200·25-s + 0.188·26-s + 0.593·29-s + 1.05·31-s + 1.01·32-s − 0.383·34-s + 0.645·37-s + 0.0776·38-s + 0.368·40-s + 0.506·41-s − 0.709·43-s − 0.392·44-s − 0.531·46-s − 0.0722·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 1.30T + 8T^{2} \) |
| 11 | \( 1 - 18.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 19.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 58.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 13.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 127.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 92.7T + 2.43e4T^{2} \) |
| 31 | \( 1 - 181.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 145.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 132.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 200.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 23.2T + 1.03e5T^{2} \) |
| 53 | \( 1 - 550.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 596.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 98.4T + 2.26e5T^{2} \) |
| 67 | \( 1 - 132.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 334.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 160.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 376.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 856.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.26e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.12e3T + 9.12e5T^{2} \) |
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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
−8.416851199005242696042747521896, −7.65169396382119450385928330425, −6.52831804383769442096733994703, −5.97815223476868533964966179326, −4.90308816527665453110144887057, −4.24735705809749637099237515180, −3.60434452260344181674385292537, −2.53579903130816107589092562255, −1.09406814599433841448236176510, 0,
1.09406814599433841448236176510, 2.53579903130816107589092562255, 3.60434452260344181674385292537, 4.24735705809749637099237515180, 4.90308816527665453110144887057, 5.97815223476868533964966179326, 6.52831804383769442096733994703, 7.65169396382119450385928330425, 8.416851199005242696042747521896
