L(s) = 1 | + 2-s − 2.36·3-s + 4-s − 2.34·5-s − 2.36·6-s + 4.14·7-s + 8-s + 2.61·9-s − 2.34·10-s + 0.755·11-s − 2.36·12-s + 5.20·13-s + 4.14·14-s + 5.54·15-s + 16-s − 0.343·17-s + 2.61·18-s + 3.14·19-s − 2.34·20-s − 9.80·21-s + 0.755·22-s + 3.00·23-s − 2.36·24-s + 0.480·25-s + 5.20·26-s + 0.915·27-s + 4.14·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.36·3-s + 0.5·4-s − 1.04·5-s − 0.967·6-s + 1.56·7-s + 0.353·8-s + 0.871·9-s − 0.740·10-s + 0.227·11-s − 0.683·12-s + 1.44·13-s + 1.10·14-s + 1.43·15-s + 0.250·16-s − 0.0833·17-s + 0.616·18-s + 0.721·19-s − 0.523·20-s − 2.14·21-s + 0.160·22-s + 0.625·23-s − 0.483·24-s + 0.0961·25-s + 1.02·26-s + 0.176·27-s + 0.782·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.034228800\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.034228800\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 2017 | \( 1 + T \) |
good | 3 | \( 1 + 2.36T + 3T^{2} \) |
| 5 | \( 1 + 2.34T + 5T^{2} \) |
| 7 | \( 1 - 4.14T + 7T^{2} \) |
| 11 | \( 1 - 0.755T + 11T^{2} \) |
| 13 | \( 1 - 5.20T + 13T^{2} \) |
| 17 | \( 1 + 0.343T + 17T^{2} \) |
| 19 | \( 1 - 3.14T + 19T^{2} \) |
| 23 | \( 1 - 3.00T + 23T^{2} \) |
| 29 | \( 1 - 0.0845T + 29T^{2} \) |
| 31 | \( 1 + 1.30T + 31T^{2} \) |
| 37 | \( 1 + 3.42T + 37T^{2} \) |
| 41 | \( 1 + 10.4T + 41T^{2} \) |
| 43 | \( 1 - 1.80T + 43T^{2} \) |
| 47 | \( 1 - 7.76T + 47T^{2} \) |
| 53 | \( 1 + 3.31T + 53T^{2} \) |
| 59 | \( 1 + 6.69T + 59T^{2} \) |
| 61 | \( 1 - 6.64T + 61T^{2} \) |
| 67 | \( 1 - 8.31T + 67T^{2} \) |
| 71 | \( 1 + 6.11T + 71T^{2} \) |
| 73 | \( 1 + 0.836T + 73T^{2} \) |
| 79 | \( 1 + 5.99T + 79T^{2} \) |
| 83 | \( 1 - 4.33T + 83T^{2} \) |
| 89 | \( 1 - 8.23T + 89T^{2} \) |
| 97 | \( 1 - 16.4T + 97T^{2} \) |
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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
−8.280891474085046200393277625049, −7.55555666893812845266205060689, −6.87586627665854227994969448346, −6.04126705662491812035446684872, −5.33523478290438634946449619401, −4.79875137975934154802335008305, −4.07289328036816783561467266160, −3.31766830049038337792593564294, −1.72308328740186053985930678356, −0.845868362028569138764444024614,
0.845868362028569138764444024614, 1.72308328740186053985930678356, 3.31766830049038337792593564294, 4.07289328036816783561467266160, 4.79875137975934154802335008305, 5.33523478290438634946449619401, 6.04126705662491812035446684872, 6.87586627665854227994969448346, 7.55555666893812845266205060689, 8.280891474085046200393277625049