| L(s) = 1 | − 3.35·3-s − 3.29·5-s − 3.21·7-s + 8.23·9-s + 2.49·11-s − 3.58·13-s + 11.0·15-s − 1.23·17-s + 19-s + 10.7·21-s + 1.16·23-s + 5.83·25-s − 17.5·27-s − 3.53·29-s + 5.21·31-s − 8.36·33-s + 10.5·35-s − 3.22·37-s + 12.0·39-s + 1.64·41-s − 6.69·43-s − 27.0·45-s − 6.93·47-s + 3.34·49-s + 4.14·51-s + 6.08·53-s − 8.22·55-s + ⋯ |
| L(s) = 1 | − 1.93·3-s − 1.47·5-s − 1.21·7-s + 2.74·9-s + 0.752·11-s − 0.995·13-s + 2.84·15-s − 0.299·17-s + 0.229·19-s + 2.35·21-s + 0.243·23-s + 1.16·25-s − 3.37·27-s − 0.657·29-s + 0.937·31-s − 1.45·33-s + 1.78·35-s − 0.530·37-s + 1.92·39-s + 0.256·41-s − 1.02·43-s − 4.03·45-s − 1.01·47-s + 0.477·49-s + 0.579·51-s + 0.836·53-s − 1.10·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.08033293695\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.08033293695\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - T \) |
| 79 | \( 1 - T \) |
| good | 3 | \( 1 + 3.35T + 3T^{2} \) |
| 5 | \( 1 + 3.29T + 5T^{2} \) |
| 7 | \( 1 + 3.21T + 7T^{2} \) |
| 11 | \( 1 - 2.49T + 11T^{2} \) |
| 13 | \( 1 + 3.58T + 13T^{2} \) |
| 17 | \( 1 + 1.23T + 17T^{2} \) |
| 23 | \( 1 - 1.16T + 23T^{2} \) |
| 29 | \( 1 + 3.53T + 29T^{2} \) |
| 31 | \( 1 - 5.21T + 31T^{2} \) |
| 37 | \( 1 + 3.22T + 37T^{2} \) |
| 41 | \( 1 - 1.64T + 41T^{2} \) |
| 43 | \( 1 + 6.69T + 43T^{2} \) |
| 47 | \( 1 + 6.93T + 47T^{2} \) |
| 53 | \( 1 - 6.08T + 53T^{2} \) |
| 59 | \( 1 + 13.2T + 59T^{2} \) |
| 61 | \( 1 + 3.04T + 61T^{2} \) |
| 67 | \( 1 + 6.29T + 67T^{2} \) |
| 71 | \( 1 - 0.128T + 71T^{2} \) |
| 73 | \( 1 + 11.6T + 73T^{2} \) |
| 83 | \( 1 + 15.6T + 83T^{2} \) |
| 89 | \( 1 + 12.0T + 89T^{2} \) |
| 97 | \( 1 - 14.5T + 97T^{2} \) |
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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
−7.69597486776755393518777957357, −7.17923771009438641915606911136, −6.63471405937495718858282588025, −6.09462259317594743232968815935, −5.16289648445177649970417470060, −4.49949472814995252406631079452, −3.91760468411407274302238104743, −3.04140040514402809829606848658, −1.38066863527220840957479698652, −0.17596944087397737767564427635,
0.17596944087397737767564427635, 1.38066863527220840957479698652, 3.04140040514402809829606848658, 3.91760468411407274302238104743, 4.49949472814995252406631079452, 5.16289648445177649970417470060, 6.09462259317594743232968815935, 6.63471405937495718858282588025, 7.17923771009438641915606911136, 7.69597486776755393518777957357
