| L(s) = 1 | + 3-s + 2.64·5-s + 4.83·7-s + 9-s − 0.990·11-s − 3.25·13-s + 2.64·15-s + 1.72·17-s + 3.60·19-s + 4.83·21-s + 5.68·23-s + 1.97·25-s + 27-s + 1.57·29-s − 10.1·31-s − 0.990·33-s + 12.7·35-s − 2.88·37-s − 3.25·39-s + 7.72·41-s + 0.980·43-s + 2.64·45-s − 13.2·47-s + 16.4·49-s + 1.72·51-s + 6.87·53-s − 2.61·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.18·5-s + 1.82·7-s + 0.333·9-s − 0.298·11-s − 0.904·13-s + 0.681·15-s + 0.417·17-s + 0.827·19-s + 1.05·21-s + 1.18·23-s + 0.395·25-s + 0.192·27-s + 0.291·29-s − 1.82·31-s − 0.172·33-s + 2.16·35-s − 0.474·37-s − 0.521·39-s + 1.20·41-s + 0.149·43-s + 0.393·45-s − 1.93·47-s + 2.34·49-s + 0.241·51-s + 0.944·53-s − 0.352·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.147589356\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.147589356\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 251 | \( 1 - T \) |
| good | 5 | \( 1 - 2.64T + 5T^{2} \) |
| 7 | \( 1 - 4.83T + 7T^{2} \) |
| 11 | \( 1 + 0.990T + 11T^{2} \) |
| 13 | \( 1 + 3.25T + 13T^{2} \) |
| 17 | \( 1 - 1.72T + 17T^{2} \) |
| 19 | \( 1 - 3.60T + 19T^{2} \) |
| 23 | \( 1 - 5.68T + 23T^{2} \) |
| 29 | \( 1 - 1.57T + 29T^{2} \) |
| 31 | \( 1 + 10.1T + 31T^{2} \) |
| 37 | \( 1 + 2.88T + 37T^{2} \) |
| 41 | \( 1 - 7.72T + 41T^{2} \) |
| 43 | \( 1 - 0.980T + 43T^{2} \) |
| 47 | \( 1 + 13.2T + 47T^{2} \) |
| 53 | \( 1 - 6.87T + 53T^{2} \) |
| 59 | \( 1 + 1.82T + 59T^{2} \) |
| 61 | \( 1 - 4.09T + 61T^{2} \) |
| 67 | \( 1 + 2.37T + 67T^{2} \) |
| 71 | \( 1 - 5.27T + 71T^{2} \) |
| 73 | \( 1 - 1.42T + 73T^{2} \) |
| 79 | \( 1 - 16.6T + 79T^{2} \) |
| 83 | \( 1 + 3.23T + 83T^{2} \) |
| 89 | \( 1 - 1.65T + 89T^{2} \) |
| 97 | \( 1 + 2.90T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.973469726084664205723309803715, −7.53031494105836213723589760479, −6.85058946839646691716370544466, −5.65564171738210850305141533958, −5.19285961485468216333924417172, −4.70264790619403936998749860236, −3.55543079696784155673693624596, −2.52471606479974534477453384414, −1.91992468295850230184085157656, −1.14981322188999275264982256576,
1.14981322188999275264982256576, 1.91992468295850230184085157656, 2.52471606479974534477453384414, 3.55543079696784155673693624596, 4.70264790619403936998749860236, 5.19285961485468216333924417172, 5.65564171738210850305141533958, 6.85058946839646691716370544466, 7.53031494105836213723589760479, 7.973469726084664205723309803715
