| L(s) = 1 | − 5.52·2-s − 4.05·3-s + 22.4·4-s + 14.0·5-s + 22.3·6-s − 79.9·8-s − 10.5·9-s − 77.8·10-s − 34.6·11-s − 91.1·12-s + 18.4·13-s − 57.1·15-s + 261.·16-s + 54.2·17-s + 58.3·18-s − 61.4·19-s + 316.·20-s + 191.·22-s + 91.8·23-s + 324.·24-s + 73.5·25-s − 102.·26-s + 152.·27-s − 7.19·29-s + 315.·30-s − 58.2·31-s − 804.·32-s + ⋯ |
| L(s) = 1 | − 1.95·2-s − 0.780·3-s + 2.81·4-s + 1.26·5-s + 1.52·6-s − 3.53·8-s − 0.391·9-s − 2.46·10-s − 0.948·11-s − 2.19·12-s + 0.394·13-s − 0.983·15-s + 4.08·16-s + 0.773·17-s + 0.764·18-s − 0.741·19-s + 3.54·20-s + 1.85·22-s + 0.832·23-s + 2.75·24-s + 0.588·25-s − 0.770·26-s + 1.08·27-s − 0.0460·29-s + 1.91·30-s − 0.337·31-s − 4.44·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.5740624659\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5740624659\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| good | 2 | \( 1 + 5.52T + 8T^{2} \) |
| 3 | \( 1 + 4.05T + 27T^{2} \) |
| 5 | \( 1 - 14.0T + 125T^{2} \) |
| 11 | \( 1 + 34.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 18.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 54.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 61.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 91.8T + 1.21e4T^{2} \) |
| 29 | \( 1 + 7.19T + 2.43e4T^{2} \) |
| 31 | \( 1 + 58.2T + 2.97e4T^{2} \) |
| 37 | \( 1 - 18.3T + 5.06e4T^{2} \) |
| 41 | \( 1 + 313.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 57.3T + 7.95e4T^{2} \) |
| 47 | \( 1 + 334.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 423.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 698.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 402.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 359.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 182.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 573.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 654.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 459.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 623.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 388.T + 9.12e5T^{2} \) |
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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
−8.653922364162920381482552078487, −8.122823459389378092917055152961, −7.10549118557772352855130563521, −6.45445111951767386510855710082, −5.75200474538272192447892802060, −5.23563217156966359919411460100, −3.17067311268077566362401614490, −2.33660823172271877237714648491, −1.46955340718946674749519606210, −0.48279940779829758414102132012,
0.48279940779829758414102132012, 1.46955340718946674749519606210, 2.33660823172271877237714648491, 3.17067311268077566362401614490, 5.23563217156966359919411460100, 5.75200474538272192447892802060, 6.45445111951767386510855710082, 7.10549118557772352855130563521, 8.122823459389378092917055152961, 8.653922364162920381482552078487
