| L(s) = 1 | − 0.128·2-s + 9.96·3-s − 7.98·4-s − 5·5-s − 1.28·6-s − 35.2·7-s + 2.05·8-s + 72.3·9-s + 0.643·10-s − 11·11-s − 79.5·12-s + 32.7·13-s + 4.53·14-s − 49.8·15-s + 63.6·16-s − 91.8·17-s − 9.31·18-s − 19·19-s + 39.9·20-s − 351.·21-s + 1.41·22-s + 163.·23-s + 20.5·24-s + 25·25-s − 4.21·26-s + 451.·27-s + 281.·28-s + ⋯ |
| L(s) = 1 | − 0.0455·2-s + 1.91·3-s − 0.997·4-s − 0.447·5-s − 0.0873·6-s − 1.90·7-s + 0.0909·8-s + 2.67·9-s + 0.0203·10-s − 0.301·11-s − 1.91·12-s + 0.698·13-s + 0.0866·14-s − 0.857·15-s + 0.993·16-s − 1.30·17-s − 0.121·18-s − 0.229·19-s + 0.446·20-s − 3.64·21-s + 0.0137·22-s + 1.47·23-s + 0.174·24-s + 0.200·25-s − 0.0317·26-s + 3.22·27-s + 1.89·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.286728711\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.286728711\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 5T \) |
| 11 | \( 1 + 11T \) |
| 19 | \( 1 + 19T \) |
| good | 2 | \( 1 + 0.128T + 8T^{2} \) |
| 3 | \( 1 - 9.96T + 27T^{2} \) |
| 7 | \( 1 + 35.2T + 343T^{2} \) |
| 13 | \( 1 - 32.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 91.8T + 4.91e3T^{2} \) |
| 23 | \( 1 - 163.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 20.1T + 2.43e4T^{2} \) |
| 31 | \( 1 - 148.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 201.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 132.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 410.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 370.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 262.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 457.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 400.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 107.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 603.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 704.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.17e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 89.0T + 5.71e5T^{2} \) |
| 89 | \( 1 - 599.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 858.T + 9.12e5T^{2} \) |
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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
−9.311656311092123361205239863062, −8.773029914660010509585290539190, −8.258792800357086875088393636887, −7.13107828924238463862504167627, −6.50014197830492382171948283605, −4.83590904907922110083037382054, −3.80527896721858727381075609486, −3.38328689241688455352497545848, −2.45572381626143403418963998372, −0.72791212504961407642773598445,
0.72791212504961407642773598445, 2.45572381626143403418963998372, 3.38328689241688455352497545848, 3.80527896721858727381075609486, 4.83590904907922110083037382054, 6.50014197830492382171948283605, 7.13107828924238463862504167627, 8.258792800357086875088393636887, 8.773029914660010509585290539190, 9.311656311092123361205239863062
