| L(s) = 1 | − 2.91·5-s + 7-s − 2·11-s − 4.35·13-s − 2.91·17-s − 19-s − 2·23-s + 3.51·25-s − 7.79·29-s + 6.35·31-s − 2.91·35-s − 3.83·37-s + 4·41-s + 3.48·43-s + 11.6·47-s + 49-s + 0.918·53-s + 5.83·55-s + 0.162·61-s + 12.7·65-s − 5.83·67-s − 0.918·71-s − 6.70·73-s − 2·77-s + 10.8·79-s − 5.79·83-s + 8.51·85-s + ⋯ |
| L(s) = 1 | − 1.30·5-s + 0.377·7-s − 0.603·11-s − 1.20·13-s − 0.707·17-s − 0.229·19-s − 0.417·23-s + 0.703·25-s − 1.44·29-s + 1.14·31-s − 0.493·35-s − 0.630·37-s + 0.624·41-s + 0.531·43-s + 1.69·47-s + 0.142·49-s + 0.126·53-s + 0.787·55-s + 0.0208·61-s + 1.57·65-s − 0.713·67-s − 0.109·71-s − 0.785·73-s − 0.227·77-s + 1.22·79-s − 0.635·83-s + 0.923·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8015815022\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8015815022\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 + 2.91T + 5T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 4.35T + 13T^{2} \) |
| 17 | \( 1 + 2.91T + 17T^{2} \) |
| 23 | \( 1 + 2T + 23T^{2} \) |
| 29 | \( 1 + 7.79T + 29T^{2} \) |
| 31 | \( 1 - 6.35T + 31T^{2} \) |
| 37 | \( 1 + 3.83T + 37T^{2} \) |
| 41 | \( 1 - 4T + 41T^{2} \) |
| 43 | \( 1 - 3.48T + 43T^{2} \) |
| 47 | \( 1 - 11.6T + 47T^{2} \) |
| 53 | \( 1 - 0.918T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 0.162T + 61T^{2} \) |
| 67 | \( 1 + 5.83T + 67T^{2} \) |
| 71 | \( 1 + 0.918T + 71T^{2} \) |
| 73 | \( 1 + 6.70T + 73T^{2} \) |
| 79 | \( 1 - 10.8T + 79T^{2} \) |
| 83 | \( 1 + 5.79T + 83T^{2} \) |
| 89 | \( 1 - 4T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−8.166228553381520396166298826976, −7.47388037568272191904455908028, −7.22457053962280901337642322414, −6.09202771701938751495185714544, −5.21459860890261353705632363201, −4.45539645557523814972599415501, −3.94205510183702123129668077054, −2.85538130259959517483978223082, −2.04936227104208245380896654922, −0.46953722464500792643323454824,
0.46953722464500792643323454824, 2.04936227104208245380896654922, 2.85538130259959517483978223082, 3.94205510183702123129668077054, 4.45539645557523814972599415501, 5.21459860890261353705632363201, 6.09202771701938751495185714544, 7.22457053962280901337642322414, 7.47388037568272191904455908028, 8.166228553381520396166298826976
