| L(s) = 1 | + 3.42·3-s + 1.19·5-s − 1.06·7-s + 8.73·9-s + 0.730·11-s + 5.49·13-s + 4.08·15-s + 0.933·17-s + 5.09·19-s − 3.63·21-s − 6.73·23-s − 3.58·25-s + 19.6·27-s + 5.95·29-s + 7.12·31-s + 2.50·33-s − 1.26·35-s + 7.45·37-s + 18.8·39-s − 11.8·41-s − 12.0·43-s + 10.4·45-s − 11.0·47-s − 5.87·49-s + 3.19·51-s − 7.95·53-s + 0.870·55-s + ⋯ |
| L(s) = 1 | + 1.97·3-s + 0.532·5-s − 0.400·7-s + 2.91·9-s + 0.220·11-s + 1.52·13-s + 1.05·15-s + 0.226·17-s + 1.16·19-s − 0.792·21-s − 1.40·23-s − 0.716·25-s + 3.78·27-s + 1.10·29-s + 1.27·31-s + 0.435·33-s − 0.213·35-s + 1.22·37-s + 3.01·39-s − 1.85·41-s − 1.83·43-s + 1.55·45-s − 1.61·47-s − 0.839·49-s + 0.447·51-s − 1.09·53-s + 0.117·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.447262838\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.447262838\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 751 | \( 1 + T \) |
| good | 3 | \( 1 - 3.42T + 3T^{2} \) |
| 5 | \( 1 - 1.19T + 5T^{2} \) |
| 7 | \( 1 + 1.06T + 7T^{2} \) |
| 11 | \( 1 - 0.730T + 11T^{2} \) |
| 13 | \( 1 - 5.49T + 13T^{2} \) |
| 17 | \( 1 - 0.933T + 17T^{2} \) |
| 19 | \( 1 - 5.09T + 19T^{2} \) |
| 23 | \( 1 + 6.73T + 23T^{2} \) |
| 29 | \( 1 - 5.95T + 29T^{2} \) |
| 31 | \( 1 - 7.12T + 31T^{2} \) |
| 37 | \( 1 - 7.45T + 37T^{2} \) |
| 41 | \( 1 + 11.8T + 41T^{2} \) |
| 43 | \( 1 + 12.0T + 43T^{2} \) |
| 47 | \( 1 + 11.0T + 47T^{2} \) |
| 53 | \( 1 + 7.95T + 53T^{2} \) |
| 59 | \( 1 + 8.66T + 59T^{2} \) |
| 61 | \( 1 - 11.6T + 61T^{2} \) |
| 67 | \( 1 - 0.0992T + 67T^{2} \) |
| 71 | \( 1 + 0.301T + 71T^{2} \) |
| 73 | \( 1 - 1.95T + 73T^{2} \) |
| 79 | \( 1 - 0.440T + 79T^{2} \) |
| 83 | \( 1 + 10.9T + 83T^{2} \) |
| 89 | \( 1 - 5.62T + 89T^{2} \) |
| 97 | \( 1 - 2.36T + 97T^{2} \) |
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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
−8.271494860291854743756696427407, −7.70244562444323186196964484010, −6.52841856052154988122858115140, −6.36820370272330466548571956431, −5.02763442848833144357577051744, −4.13792283542897478167010776906, −3.36473528685962110368444472153, −3.01510179532937115545081123924, −1.86781538172883336085712541321, −1.29557962209679009276266506603,
1.29557962209679009276266506603, 1.86781538172883336085712541321, 3.01510179532937115545081123924, 3.36473528685962110368444472153, 4.13792283542897478167010776906, 5.02763442848833144357577051744, 6.36820370272330466548571956431, 6.52841856052154988122858115140, 7.70244562444323186196964484010, 8.271494860291854743756696427407
