L(s) = 1 | + 1.38·3-s − 3.43·5-s + 3.30·7-s − 1.09·9-s − 0.715·11-s + 5.15·13-s − 4.73·15-s − 1.52·17-s + 7.44·19-s + 4.56·21-s − 3.93·23-s + 6.78·25-s − 5.65·27-s − 4.20·29-s + 6.78·31-s − 0.987·33-s − 11.3·35-s − 5.07·37-s + 7.11·39-s + 1.57·41-s − 1.14·43-s + 3.75·45-s + 4.06·47-s + 3.93·49-s − 2.10·51-s + 3.64·53-s + 2.45·55-s + ⋯ |
L(s) = 1 | + 0.796·3-s − 1.53·5-s + 1.24·7-s − 0.364·9-s − 0.215·11-s + 1.42·13-s − 1.22·15-s − 0.370·17-s + 1.70·19-s + 0.996·21-s − 0.821·23-s + 1.35·25-s − 1.08·27-s − 0.780·29-s + 1.21·31-s − 0.171·33-s − 1.91·35-s − 0.835·37-s + 1.13·39-s + 0.245·41-s − 0.174·43-s + 0.560·45-s + 0.593·47-s + 0.562·49-s − 0.295·51-s + 0.500·53-s + 0.331·55-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
\(L(1)\) |
\(\approx\) |
\(2.261951872\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.261951872\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.38T + 3T^{2} \) |
| 5 | \( 1 + 3.43T + 5T^{2} \) |
| 7 | \( 1 - 3.30T + 7T^{2} \) |
| 11 | \( 1 + 0.715T + 11T^{2} \) |
| 13 | \( 1 - 5.15T + 13T^{2} \) |
| 17 | \( 1 + 1.52T + 17T^{2} \) |
| 19 | \( 1 - 7.44T + 19T^{2} \) |
| 23 | \( 1 + 3.93T + 23T^{2} \) |
| 29 | \( 1 + 4.20T + 29T^{2} \) |
| 31 | \( 1 - 6.78T + 31T^{2} \) |
| 37 | \( 1 + 5.07T + 37T^{2} \) |
| 41 | \( 1 - 1.57T + 41T^{2} \) |
| 43 | \( 1 + 1.14T + 43T^{2} \) |
| 47 | \( 1 - 4.06T + 47T^{2} \) |
| 53 | \( 1 - 3.64T + 53T^{2} \) |
| 59 | \( 1 - 12.5T + 59T^{2} \) |
| 61 | \( 1 + 5.54T + 61T^{2} \) |
| 67 | \( 1 + 2.31T + 67T^{2} \) |
| 71 | \( 1 - 6.03T + 71T^{2} \) |
| 73 | \( 1 + 6.14T + 73T^{2} \) |
| 79 | \( 1 - 12.9T + 79T^{2} \) |
| 83 | \( 1 - 8.06T + 83T^{2} \) |
| 89 | \( 1 + 15.5T + 89T^{2} \) |
| 97 | \( 1 - 4.13T + 97T^{2} \) |
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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
−8.000755216097491021909777584786, −7.76116608032384691395612733673, −6.95568309599235798775109093024, −5.83517456939965704922926994396, −5.11815728588047127479625725280, −4.19709604131241306281598785938, −3.66884404641986038490317098069, −2.98712294507398582787980550506, −1.86864412156190462960282737316, −0.77855816035910049001272546494,
0.77855816035910049001272546494, 1.86864412156190462960282737316, 2.98712294507398582787980550506, 3.66884404641986038490317098069, 4.19709604131241306281598785938, 5.11815728588047127479625725280, 5.83517456939965704922926994396, 6.95568309599235798775109093024, 7.76116608032384691395612733673, 8.000755216097491021909777584786