| L(s) = 1 | − 2-s + 4-s + 5-s − 8-s − 10-s + 5·11-s + 5·13-s + 16-s − 4·17-s + 7·19-s + 20-s − 5·22-s − 23-s + 25-s − 5·26-s + 2·31-s − 32-s + 4·34-s + 37-s − 7·38-s − 40-s + 5·41-s + 12·43-s + 5·44-s + 46-s − 11·47-s − 50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.353·8-s − 0.316·10-s + 1.50·11-s + 1.38·13-s + 1/4·16-s − 0.970·17-s + 1.60·19-s + 0.223·20-s − 1.06·22-s − 0.208·23-s + 1/5·25-s − 0.980·26-s + 0.359·31-s − 0.176·32-s + 0.685·34-s + 0.164·37-s − 1.13·38-s − 0.158·40-s + 0.780·41-s + 1.82·43-s + 0.753·44-s + 0.147·46-s − 1.60·47-s − 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.933158929\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.933158929\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 11 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−8.670774440110314087997085164121, −7.61570821190173541461733422101, −6.96041805233112958138529978091, −6.16582625079179459302280288733, −5.79253797329377784369369211934, −4.51332993285861250424347613881, −3.72890789033593310275662800766, −2.81209690346494303693916389029, −1.62419940440162722351245363466, −0.964950264048313875888571738404,
0.964950264048313875888571738404, 1.62419940440162722351245363466, 2.81209690346494303693916389029, 3.72890789033593310275662800766, 4.51332993285861250424347613881, 5.79253797329377784369369211934, 6.16582625079179459302280288733, 6.96041805233112958138529978091, 7.61570821190173541461733422101, 8.670774440110314087997085164121
