| L(s) = 1 | + 3-s + 3.25·5-s − 1.95·7-s + 9-s − 1.86·11-s + 4.83·13-s + 3.25·15-s − 4.17·17-s − 5.05·19-s − 1.95·21-s + 1.26·23-s + 5.62·25-s + 27-s + 10.5·29-s + 6.26·31-s − 1.86·33-s − 6.38·35-s + 0.409·37-s + 4.83·39-s − 2.65·41-s − 0.317·43-s + 3.25·45-s + 4.08·47-s − 3.16·49-s − 4.17·51-s − 7.15·53-s − 6.08·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.45·5-s − 0.740·7-s + 0.333·9-s − 0.562·11-s + 1.34·13-s + 0.841·15-s − 1.01·17-s − 1.15·19-s − 0.427·21-s + 0.262·23-s + 1.12·25-s + 0.192·27-s + 1.95·29-s + 1.12·31-s − 0.324·33-s − 1.07·35-s + 0.0673·37-s + 0.774·39-s − 0.414·41-s − 0.0483·43-s + 0.485·45-s + 0.596·47-s − 0.451·49-s − 0.584·51-s − 0.983·53-s − 0.820·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.285540484\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.285540484\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 167 | \( 1 - T \) |
| good | 5 | \( 1 - 3.25T + 5T^{2} \) |
| 7 | \( 1 + 1.95T + 7T^{2} \) |
| 11 | \( 1 + 1.86T + 11T^{2} \) |
| 13 | \( 1 - 4.83T + 13T^{2} \) |
| 17 | \( 1 + 4.17T + 17T^{2} \) |
| 19 | \( 1 + 5.05T + 19T^{2} \) |
| 23 | \( 1 - 1.26T + 23T^{2} \) |
| 29 | \( 1 - 10.5T + 29T^{2} \) |
| 31 | \( 1 - 6.26T + 31T^{2} \) |
| 37 | \( 1 - 0.409T + 37T^{2} \) |
| 41 | \( 1 + 2.65T + 41T^{2} \) |
| 43 | \( 1 + 0.317T + 43T^{2} \) |
| 47 | \( 1 - 4.08T + 47T^{2} \) |
| 53 | \( 1 + 7.15T + 53T^{2} \) |
| 59 | \( 1 - 8.72T + 59T^{2} \) |
| 61 | \( 1 - 9.89T + 61T^{2} \) |
| 67 | \( 1 - 6.55T + 67T^{2} \) |
| 71 | \( 1 - 10.0T + 71T^{2} \) |
| 73 | \( 1 + 11.8T + 73T^{2} \) |
| 79 | \( 1 - 2.70T + 79T^{2} \) |
| 83 | \( 1 + 6.18T + 83T^{2} \) |
| 89 | \( 1 - 14.6T + 89T^{2} \) |
| 97 | \( 1 - 10.3T + 97T^{2} \) |
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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
−8.097998920459137171858539374008, −6.79481466924129854708728397511, −6.47962865004968855729191506222, −5.99440227229306837847710839152, −5.02747042668430433414806965977, −4.29995416972713780555512761525, −3.32475235038047807426092261128, −2.56791327691311508505522298675, −1.99002614776947549996829895814, −0.878404481983693771776190993166,
0.878404481983693771776190993166, 1.99002614776947549996829895814, 2.56791327691311508505522298675, 3.32475235038047807426092261128, 4.29995416972713780555512761525, 5.02747042668430433414806965977, 5.99440227229306837847710839152, 6.47962865004968855729191506222, 6.79481466924129854708728397511, 8.097998920459137171858539374008
