| L(s) = 1 | − 3.68·3-s − 0.561·5-s − 18.1·7-s − 13.4·9-s + 64.7·11-s + 13·13-s + 2.06·15-s − 25.5·17-s − 107.·19-s + 66.9·21-s − 73.2·23-s − 124.·25-s + 148.·27-s − 175.·29-s + 113.·31-s − 238.·33-s + 10.2·35-s − 114.·37-s − 47.9·39-s − 69.6·41-s + 438.·43-s + 7.53·45-s + 31.9·47-s − 12.5·49-s + 94.1·51-s − 2.84·53-s − 36.3·55-s + ⋯ |
| L(s) = 1 | − 0.709·3-s − 0.0502·5-s − 0.981·7-s − 0.497·9-s + 1.77·11-s + 0.277·13-s + 0.0356·15-s − 0.364·17-s − 1.30·19-s + 0.695·21-s − 0.664·23-s − 0.997·25-s + 1.06·27-s − 1.12·29-s + 0.655·31-s − 1.25·33-s + 0.0492·35-s − 0.510·37-s − 0.196·39-s − 0.265·41-s + 1.55·43-s + 0.0249·45-s + 0.0991·47-s − 0.0367·49-s + 0.258·51-s − 0.00737·53-s − 0.0891·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9731259226\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9731259226\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 - 13T \) |
| good | 3 | \( 1 + 3.68T + 27T^{2} \) |
| 5 | \( 1 + 0.561T + 125T^{2} \) |
| 7 | \( 1 + 18.1T + 343T^{2} \) |
| 11 | \( 1 - 64.7T + 1.33e3T^{2} \) |
| 17 | \( 1 + 25.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 107.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 73.2T + 1.21e4T^{2} \) |
| 29 | \( 1 + 175.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 113.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 114.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 69.6T + 6.89e4T^{2} \) |
| 43 | \( 1 - 438.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 31.9T + 1.03e5T^{2} \) |
| 53 | \( 1 + 2.84T + 1.48e5T^{2} \) |
| 59 | \( 1 - 71.6T + 2.05e5T^{2} \) |
| 61 | \( 1 - 920.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 444.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 541.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 764.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 421.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 603.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.15e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 583.T + 9.12e5T^{2} \) |
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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
−9.763049662525446395019995029923, −9.087716683540676492464833672287, −8.259484174151348598214097865757, −6.87647617780041753909116587389, −6.32278332797021186045669683889, −5.71031202655987873146794101578, −4.26840125104223948878269564929, −3.57017249181654332997882054386, −2.07823039899284896462840344457, −0.55174718232460916016725526843,
0.55174718232460916016725526843, 2.07823039899284896462840344457, 3.57017249181654332997882054386, 4.26840125104223948878269564929, 5.71031202655987873146794101578, 6.32278332797021186045669683889, 6.87647617780041753909116587389, 8.259484174151348598214097865757, 9.087716683540676492464833672287, 9.763049662525446395019995029923
