| L(s) = 1 | + 5.38·2-s + 20.9·4-s − 5·5-s + 12.5·7-s + 69.9·8-s − 26.9·10-s + 12.8·11-s + 59.8·13-s + 67.6·14-s + 208.·16-s − 110.·17-s − 12.0·19-s − 104.·20-s + 69.0·22-s − 67.7·23-s + 25·25-s + 322.·26-s + 263.·28-s + 199.·29-s + 76.6·31-s + 565.·32-s − 592.·34-s − 62.8·35-s − 22.4·37-s − 65.0·38-s − 349.·40-s + 87.7·41-s + ⋯ |
| L(s) = 1 | + 1.90·2-s + 2.62·4-s − 0.447·5-s + 0.678·7-s + 3.09·8-s − 0.851·10-s + 0.351·11-s + 1.27·13-s + 1.29·14-s + 3.26·16-s − 1.56·17-s − 0.145·19-s − 1.17·20-s + 0.669·22-s − 0.613·23-s + 0.200·25-s + 2.43·26-s + 1.78·28-s + 1.28·29-s + 0.443·31-s + 3.12·32-s − 2.98·34-s − 0.303·35-s − 0.0998·37-s − 0.277·38-s − 1.38·40-s + 0.334·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(6.635080989\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.635080989\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| good | 2 | \( 1 - 5.38T + 8T^{2} \) |
| 7 | \( 1 - 12.5T + 343T^{2} \) |
| 11 | \( 1 - 12.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 59.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 110.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 12.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + 67.7T + 1.21e4T^{2} \) |
| 29 | \( 1 - 199.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 76.6T + 2.97e4T^{2} \) |
| 37 | \( 1 + 22.4T + 5.06e4T^{2} \) |
| 41 | \( 1 - 87.7T + 6.89e4T^{2} \) |
| 43 | \( 1 - 119.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 243.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 293.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 581.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 773.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 231.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 744.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 264.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 559.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.22e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 255.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.04e3T + 9.12e5T^{2} \) |
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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
−11.16367822216126260023413695974, −10.59374581461984949794635850555, −8.772539296728125686840746225472, −7.75568253154504959856833798974, −6.61323815346549005315335532433, −5.98832554641626732896442875770, −4.63407213640431552555192754310, −4.16463209098446068639553574114, −2.93701824830953775772850386143, −1.60077714316732582054481276997,
1.60077714316732582054481276997, 2.93701824830953775772850386143, 4.16463209098446068639553574114, 4.63407213640431552555192754310, 5.98832554641626732896442875770, 6.61323815346549005315335532433, 7.75568253154504959856833798974, 8.772539296728125686840746225472, 10.59374581461984949794635850555, 11.16367822216126260023413695974
