| L(s) = 1 | − 2-s + 4-s − 5-s + 3.37·7-s − 8-s + 10-s − 0.627·11-s − 2·13-s − 3.37·14-s + 16-s + 4.74·17-s − 0.627·19-s − 20-s + 0.627·22-s − 3.37·23-s + 25-s + 2·26-s + 3.37·28-s + 8.74·29-s − 31-s − 32-s − 4.74·34-s − 3.37·35-s − 0.744·37-s + 0.627·38-s + 40-s − 0.744·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.447·5-s + 1.27·7-s − 0.353·8-s + 0.316·10-s − 0.189·11-s − 0.554·13-s − 0.901·14-s + 0.250·16-s + 1.15·17-s − 0.144·19-s − 0.223·20-s + 0.133·22-s − 0.703·23-s + 0.200·25-s + 0.392·26-s + 0.637·28-s + 1.62·29-s − 0.179·31-s − 0.176·32-s − 0.813·34-s − 0.570·35-s − 0.122·37-s + 0.101·38-s + 0.158·40-s − 0.116·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.364583800\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.364583800\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| good | 7 | \( 1 - 3.37T + 7T^{2} \) |
| 11 | \( 1 + 0.627T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 - 4.74T + 17T^{2} \) |
| 19 | \( 1 + 0.627T + 19T^{2} \) |
| 23 | \( 1 + 3.37T + 23T^{2} \) |
| 29 | \( 1 - 8.74T + 29T^{2} \) |
| 37 | \( 1 + 0.744T + 37T^{2} \) |
| 41 | \( 1 + 0.744T + 41T^{2} \) |
| 43 | \( 1 - 0.627T + 43T^{2} \) |
| 47 | \( 1 - 6.74T + 47T^{2} \) |
| 53 | \( 1 + 1.37T + 53T^{2} \) |
| 59 | \( 1 + 2.74T + 59T^{2} \) |
| 61 | \( 1 - 11.4T + 61T^{2} \) |
| 67 | \( 1 - 10.7T + 67T^{2} \) |
| 71 | \( 1 + 3.37T + 71T^{2} \) |
| 73 | \( 1 + 8.11T + 73T^{2} \) |
| 79 | \( 1 + 4.62T + 79T^{2} \) |
| 83 | \( 1 - 12T + 83T^{2} \) |
| 89 | \( 1 - 1.37T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−8.558194309431281927996548678523, −8.100014848148849208612908958553, −7.55694201394260437153512967792, −6.77175869783683431060238249988, −5.69602831886658765301387807643, −4.94901421930186094263349090598, −4.08505674489598037921967125179, −2.93291269551635920680915396404, −1.92147104150213742762645257923, −0.826941842145937186901749736447,
0.826941842145937186901749736447, 1.92147104150213742762645257923, 2.93291269551635920680915396404, 4.08505674489598037921967125179, 4.94901421930186094263349090598, 5.69602831886658765301387807643, 6.77175869783683431060238249988, 7.55694201394260437153512967792, 8.100014848148849208612908958553, 8.558194309431281927996548678523
