| L(s) = 1 | − 3-s − 4.37·5-s + 0.0538·7-s + 9-s + 1.79·11-s + 3.12·13-s + 4.37·15-s + 4.85·17-s − 3.63·19-s − 0.0538·21-s + 8.81·23-s + 14.1·25-s − 27-s − 4.82·29-s − 4.67·31-s − 1.79·33-s − 0.235·35-s − 3.90·37-s − 3.12·39-s − 9.25·41-s − 2.09·43-s − 4.37·45-s + 5.77·47-s − 6.99·49-s − 4.85·51-s + 12.6·53-s − 7.87·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.95·5-s + 0.0203·7-s + 0.333·9-s + 0.542·11-s + 0.866·13-s + 1.13·15-s + 1.17·17-s − 0.832·19-s − 0.0117·21-s + 1.83·23-s + 2.83·25-s − 0.192·27-s − 0.895·29-s − 0.839·31-s − 0.313·33-s − 0.0398·35-s − 0.641·37-s − 0.500·39-s − 1.44·41-s − 0.319·43-s − 0.652·45-s + 0.842·47-s − 0.999·49-s − 0.679·51-s + 1.73·53-s − 1.06·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6096 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9531375822\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9531375822\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 127 | \( 1 - T \) |
| good | 5 | \( 1 + 4.37T + 5T^{2} \) |
| 7 | \( 1 - 0.0538T + 7T^{2} \) |
| 11 | \( 1 - 1.79T + 11T^{2} \) |
| 13 | \( 1 - 3.12T + 13T^{2} \) |
| 17 | \( 1 - 4.85T + 17T^{2} \) |
| 19 | \( 1 + 3.63T + 19T^{2} \) |
| 23 | \( 1 - 8.81T + 23T^{2} \) |
| 29 | \( 1 + 4.82T + 29T^{2} \) |
| 31 | \( 1 + 4.67T + 31T^{2} \) |
| 37 | \( 1 + 3.90T + 37T^{2} \) |
| 41 | \( 1 + 9.25T + 41T^{2} \) |
| 43 | \( 1 + 2.09T + 43T^{2} \) |
| 47 | \( 1 - 5.77T + 47T^{2} \) |
| 53 | \( 1 - 12.6T + 53T^{2} \) |
| 59 | \( 1 + 12.6T + 59T^{2} \) |
| 61 | \( 1 + 13.2T + 61T^{2} \) |
| 67 | \( 1 - 2.44T + 67T^{2} \) |
| 71 | \( 1 - 12.0T + 71T^{2} \) |
| 73 | \( 1 + 12.8T + 73T^{2} \) |
| 79 | \( 1 - 7.31T + 79T^{2} \) |
| 83 | \( 1 - 4.17T + 83T^{2} \) |
| 89 | \( 1 + 12.5T + 89T^{2} \) |
| 97 | \( 1 - 1.30T + 97T^{2} \) |
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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
−8.004238180635029932238130133690, −7.25804640753871249896708370099, −6.89686231846011508324530363602, −5.95221148949582688899029559177, −5.06188815171170727510372375126, −4.41797531897129051741637480384, −3.54131047652222894236411156347, −3.28243717224608348377582638124, −1.52189783386475620864168461631, −0.56185509856214868675832866961,
0.56185509856214868675832866961, 1.52189783386475620864168461631, 3.28243717224608348377582638124, 3.54131047652222894236411156347, 4.41797531897129051741637480384, 5.06188815171170727510372375126, 5.95221148949582688899029559177, 6.89686231846011508324530363602, 7.25804640753871249896708370099, 8.004238180635029932238130133690
