L(s) = 1 | + 4·2-s + 30.5·3-s + 16·4-s − 25·5-s + 122.·6-s − 49·7-s + 64·8-s + 688.·9-s − 100·10-s + 392.·11-s + 488.·12-s − 631.·13-s − 196·14-s − 763.·15-s + 256·16-s − 1.37e3·17-s + 2.75e3·18-s + 1.49e3·19-s − 400·20-s − 1.49e3·21-s + 1.56e3·22-s − 4.57e3·23-s + 1.95e3·24-s + 625·25-s − 2.52e3·26-s + 1.35e4·27-s − 784·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.95·3-s + 0.5·4-s − 0.447·5-s + 1.38·6-s − 0.377·7-s + 0.353·8-s + 2.83·9-s − 0.316·10-s + 0.977·11-s + 0.978·12-s − 1.03·13-s − 0.267·14-s − 0.875·15-s + 0.250·16-s − 1.15·17-s + 2.00·18-s + 0.951·19-s − 0.223·20-s − 0.740·21-s + 0.691·22-s − 1.80·23-s + 0.692·24-s + 0.200·25-s − 0.733·26-s + 3.59·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(4.455164068\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.455164068\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 4T \) |
| 5 | \( 1 + 25T \) |
| 7 | \( 1 + 49T \) |
good | 3 | \( 1 - 30.5T + 243T^{2} \) |
| 11 | \( 1 - 392.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 631.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.37e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.49e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 4.57e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.70e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 6.93e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.47e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.47e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.07e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 6.47e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.26e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.92e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.64e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 828.T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.80e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 7.61e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.07e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 9.40e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 4.35e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 3.41e4T + 8.58e9T^{2} \) |
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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
−13.94889412934630709668902917304, −12.92192468627101300965225775728, −11.88806597068515224093009565664, −10.02108248265011488039728458141, −9.057862781152959972362994168154, −7.80795271579086658016837742813, −6.81345856429996430442023192835, −4.38709060751967225066505483538, −3.39080156827967761729995215346, −2.04716824787560649346165555337,
2.04716824787560649346165555337, 3.39080156827967761729995215346, 4.38709060751967225066505483538, 6.81345856429996430442023192835, 7.80795271579086658016837742813, 9.057862781152959972362994168154, 10.02108248265011488039728458141, 11.88806597068515224093009565664, 12.92192468627101300965225775728, 13.94889412934630709668902917304