| L(s) = 1 | − 0.446·3-s + 5-s + 3.25·7-s − 2.80·9-s − 1.99·11-s − 3.05·13-s − 0.446·15-s + 8.07·17-s + 2.16·19-s − 1.45·21-s − 4.14·23-s + 25-s + 2.58·27-s + 4.82·29-s − 3.36·31-s + 0.889·33-s + 3.25·35-s − 1.82·37-s + 1.36·39-s + 7.04·41-s + 6.10·43-s − 2.80·45-s − 3.36·47-s + 3.61·49-s − 3.60·51-s − 1.14·53-s − 1.99·55-s + ⋯ |
| L(s) = 1 | − 0.257·3-s + 0.447·5-s + 1.23·7-s − 0.933·9-s − 0.600·11-s − 0.847·13-s − 0.115·15-s + 1.95·17-s + 0.497·19-s − 0.317·21-s − 0.864·23-s + 0.200·25-s + 0.498·27-s + 0.895·29-s − 0.603·31-s + 0.154·33-s + 0.550·35-s − 0.299·37-s + 0.218·39-s + 1.09·41-s + 0.931·43-s − 0.417·45-s − 0.490·47-s + 0.515·49-s − 0.504·51-s − 0.157·53-s − 0.268·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.103583617\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.103583617\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 - T \) |
| good | 3 | \( 1 + 0.446T + 3T^{2} \) |
| 7 | \( 1 - 3.25T + 7T^{2} \) |
| 11 | \( 1 + 1.99T + 11T^{2} \) |
| 13 | \( 1 + 3.05T + 13T^{2} \) |
| 17 | \( 1 - 8.07T + 17T^{2} \) |
| 19 | \( 1 - 2.16T + 19T^{2} \) |
| 23 | \( 1 + 4.14T + 23T^{2} \) |
| 29 | \( 1 - 4.82T + 29T^{2} \) |
| 31 | \( 1 + 3.36T + 31T^{2} \) |
| 37 | \( 1 + 1.82T + 37T^{2} \) |
| 41 | \( 1 - 7.04T + 41T^{2} \) |
| 43 | \( 1 - 6.10T + 43T^{2} \) |
| 47 | \( 1 + 3.36T + 47T^{2} \) |
| 53 | \( 1 + 1.14T + 53T^{2} \) |
| 59 | \( 1 + 8.30T + 59T^{2} \) |
| 61 | \( 1 + 3.17T + 61T^{2} \) |
| 67 | \( 1 + 3.60T + 67T^{2} \) |
| 71 | \( 1 - 3.46T + 71T^{2} \) |
| 73 | \( 1 - 2.96T + 73T^{2} \) |
| 79 | \( 1 - 13.1T + 79T^{2} \) |
| 83 | \( 1 - 1.13T + 83T^{2} \) |
| 89 | \( 1 - 12.5T + 89T^{2} \) |
| 97 | \( 1 + 6.31T + 97T^{2} \) |
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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
−7.86545675018587540128225118825, −7.38660070003467316878047486789, −6.24684527731475312355380872929, −5.59450819524946364979123785435, −5.19133490231099317115950568501, −4.53967366442304039225073000587, −3.37082503640470099171819133885, −2.63738739354068859792112529085, −1.77489579755534191996515831389, −0.73663746884936712164134085077,
0.73663746884936712164134085077, 1.77489579755534191996515831389, 2.63738739354068859792112529085, 3.37082503640470099171819133885, 4.53967366442304039225073000587, 5.19133490231099317115950568501, 5.59450819524946364979123785435, 6.24684527731475312355380872929, 7.38660070003467316878047486789, 7.86545675018587540128225118825
