L(s) = 1 | + 3-s + 0.169·5-s + 1.46·7-s + 9-s + 11-s + 13-s + 0.169·15-s + 6.72·17-s + 6.80·19-s + 1.46·21-s − 5.29·23-s − 4.97·25-s + 27-s − 4.23·29-s + 7.17·31-s + 33-s + 0.247·35-s + 5.03·37-s + 39-s + 0.720·41-s − 12.1·43-s + 0.169·45-s + 10.4·47-s − 4.86·49-s + 6.72·51-s + 12.9·53-s + 0.169·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.0757·5-s + 0.552·7-s + 0.333·9-s + 0.301·11-s + 0.277·13-s + 0.0437·15-s + 1.63·17-s + 1.56·19-s + 0.319·21-s − 1.10·23-s − 0.994·25-s + 0.192·27-s − 0.785·29-s + 1.28·31-s + 0.174·33-s + 0.0418·35-s + 0.826·37-s + 0.160·39-s + 0.112·41-s − 1.85·43-s + 0.0252·45-s + 1.52·47-s − 0.694·49-s + 0.941·51-s + 1.77·53-s + 0.0228·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.259538123\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.259538123\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
good | 5 | \( 1 - 0.169T + 5T^{2} \) |
| 7 | \( 1 - 1.46T + 7T^{2} \) |
| 17 | \( 1 - 6.72T + 17T^{2} \) |
| 19 | \( 1 - 6.80T + 19T^{2} \) |
| 23 | \( 1 + 5.29T + 23T^{2} \) |
| 29 | \( 1 + 4.23T + 29T^{2} \) |
| 31 | \( 1 - 7.17T + 31T^{2} \) |
| 37 | \( 1 - 5.03T + 37T^{2} \) |
| 41 | \( 1 - 0.720T + 41T^{2} \) |
| 43 | \( 1 + 12.1T + 43T^{2} \) |
| 47 | \( 1 - 10.4T + 47T^{2} \) |
| 53 | \( 1 - 12.9T + 53T^{2} \) |
| 59 | \( 1 + 3.92T + 59T^{2} \) |
| 61 | \( 1 - 1.23T + 61T^{2} \) |
| 67 | \( 1 + 0.504T + 67T^{2} \) |
| 71 | \( 1 - 12.0T + 71T^{2} \) |
| 73 | \( 1 + 3.12T + 73T^{2} \) |
| 79 | \( 1 + 6.05T + 79T^{2} \) |
| 83 | \( 1 - 16.0T + 83T^{2} \) |
| 89 | \( 1 + 1.80T + 89T^{2} \) |
| 97 | \( 1 + 9.83T + 97T^{2} \) |
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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
−7.916418455636406128949142250716, −7.54967524772402683349835054316, −6.62214337995571700474112645972, −5.70392716650886999355986641355, −5.27351816880945719861240226081, −4.17959298896063715097702653429, −3.60155283944887728569939878866, −2.78610223433225189142919449863, −1.76006028555787896386126410809, −0.968665162519282270619096063035,
0.968665162519282270619096063035, 1.76006028555787896386126410809, 2.78610223433225189142919449863, 3.60155283944887728569939878866, 4.17959298896063715097702653429, 5.27351816880945719861240226081, 5.70392716650886999355986641355, 6.62214337995571700474112645972, 7.54967524772402683349835054316, 7.916418455636406128949142250716