| L(s) = 1 | + 2·2-s + 4·4-s − 19.0·5-s − 26.4·7-s + 8·8-s − 38.0·10-s + 27.2·11-s + 69.8·13-s − 52.8·14-s + 16·16-s − 15.2·17-s + 65.6·19-s − 76.0·20-s + 54.5·22-s + 23·23-s + 236.·25-s + 139.·26-s − 105.·28-s + 124.·29-s + 302.·31-s + 32·32-s − 30.4·34-s + 501.·35-s + 5.93·37-s + 131.·38-s − 152.·40-s − 166.·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.70·5-s − 1.42·7-s + 0.353·8-s − 1.20·10-s + 0.747·11-s + 1.49·13-s − 1.00·14-s + 0.250·16-s − 0.217·17-s + 0.792·19-s − 0.850·20-s + 0.528·22-s + 0.208·23-s + 1.89·25-s + 1.05·26-s − 0.712·28-s + 0.797·29-s + 1.75·31-s + 0.176·32-s − 0.153·34-s + 2.42·35-s + 0.0263·37-s + 0.560·38-s − 0.601·40-s − 0.634·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.946406699\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.946406699\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 \) |
| 23 | \( 1 - 23T \) |
| good | 5 | \( 1 + 19.0T + 125T^{2} \) |
| 7 | \( 1 + 26.4T + 343T^{2} \) |
| 11 | \( 1 - 27.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 69.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 15.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 65.6T + 6.85e3T^{2} \) |
| 29 | \( 1 - 124.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 302.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 5.93T + 5.06e4T^{2} \) |
| 41 | \( 1 + 166.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 318.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 144.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 502.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 600.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 636.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 206.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 440.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 390.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 681.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 695.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 948.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 746.T + 9.12e5T^{2} \) |
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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
−11.14044527217528452663126904146, −10.02995796253366823227421603309, −8.829253789730019435628786545438, −7.962869098206419683044557819443, −6.77595756713151111035537033644, −6.29097252415773959660569463237, −4.67945145748774131952369745174, −3.63169375537697140880855014622, −3.20194076589433958526125914987, −0.823706817954258079005968969562,
0.823706817954258079005968969562, 3.20194076589433958526125914987, 3.63169375537697140880855014622, 4.67945145748774131952369745174, 6.29097252415773959660569463237, 6.77595756713151111035537033644, 7.962869098206419683044557819443, 8.829253789730019435628786545438, 10.02995796253366823227421603309, 11.14044527217528452663126904146
