| L(s) = 1 | − 1.21·2-s + 1.31·3-s − 0.525·4-s − 1.59·6-s − 7-s + 3.06·8-s − 1.28·9-s + 11-s − 0.688·12-s + 2.68·13-s + 1.21·14-s − 2.67·16-s + 4.68·17-s + 1.55·18-s − 8.23·19-s − 1.31·21-s − 1.21·22-s + 4.14·23-s + 4.02·24-s − 3.26·26-s − 5.61·27-s + 0.525·28-s + 5.05·29-s + 5.39·31-s − 2.88·32-s + 1.31·33-s − 5.69·34-s + ⋯ |
| L(s) = 1 | − 0.858·2-s + 0.756·3-s − 0.262·4-s − 0.649·6-s − 0.377·7-s + 1.08·8-s − 0.426·9-s + 0.301·11-s − 0.198·12-s + 0.745·13-s + 0.324·14-s − 0.668·16-s + 1.13·17-s + 0.366·18-s − 1.88·19-s − 0.286·21-s − 0.258·22-s + 0.864·23-s + 0.820·24-s − 0.640·26-s − 1.08·27-s + 0.0992·28-s + 0.937·29-s + 0.969·31-s − 0.510·32-s + 0.228·33-s − 0.976·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.159477114\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.159477114\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 + 1.21T + 2T^{2} \) |
| 3 | \( 1 - 1.31T + 3T^{2} \) |
| 13 | \( 1 - 2.68T + 13T^{2} \) |
| 17 | \( 1 - 4.68T + 17T^{2} \) |
| 19 | \( 1 + 8.23T + 19T^{2} \) |
| 23 | \( 1 - 4.14T + 23T^{2} \) |
| 29 | \( 1 - 5.05T + 29T^{2} \) |
| 31 | \( 1 - 5.39T + 31T^{2} \) |
| 37 | \( 1 + 4.76T + 37T^{2} \) |
| 41 | \( 1 + 1.16T + 41T^{2} \) |
| 43 | \( 1 + 5.95T + 43T^{2} \) |
| 47 | \( 1 - 8.68T + 47T^{2} \) |
| 53 | \( 1 - 0.769T + 53T^{2} \) |
| 59 | \( 1 - 4.02T + 59T^{2} \) |
| 61 | \( 1 + 0.407T + 61T^{2} \) |
| 67 | \( 1 - 13.1T + 67T^{2} \) |
| 71 | \( 1 + 2.19T + 71T^{2} \) |
| 73 | \( 1 - 10.9T + 73T^{2} \) |
| 79 | \( 1 - 8.51T + 79T^{2} \) |
| 83 | \( 1 - 9.90T + 83T^{2} \) |
| 89 | \( 1 + 9.80T + 89T^{2} \) |
| 97 | \( 1 + 8.85T + 97T^{2} \) |
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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
−9.008527849894461732125898487081, −8.457253259428265221565611507623, −8.117666516766333525721492790929, −7.01195447033711804641589824575, −6.21815698727878346777447876267, −5.12401955621082479024244046347, −4.06286141668177964051079698120, −3.27037932166590933701054706922, −2.11141693067647682838687238249, −0.808195178823484348223054022867,
0.808195178823484348223054022867, 2.11141693067647682838687238249, 3.27037932166590933701054706922, 4.06286141668177964051079698120, 5.12401955621082479024244046347, 6.21815698727878346777447876267, 7.01195447033711804641589824575, 8.117666516766333525721492790929, 8.457253259428265221565611507623, 9.008527849894461732125898487081
