| L(s) = 1 | + 2-s + 4-s + 1.20·5-s + 4.22·7-s + 8-s + 1.20·10-s + 1.67·11-s − 3.39·13-s + 4.22·14-s + 16-s + 6.30·17-s + 0.549·19-s + 1.20·20-s + 1.67·22-s − 1.62·23-s − 3.54·25-s − 3.39·26-s + 4.22·28-s − 5.50·29-s + 5.74·31-s + 32-s + 6.30·34-s + 5.08·35-s + 6.81·37-s + 0.549·38-s + 1.20·40-s − 1.19·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.538·5-s + 1.59·7-s + 0.353·8-s + 0.381·10-s + 0.506·11-s − 0.941·13-s + 1.12·14-s + 0.250·16-s + 1.52·17-s + 0.126·19-s + 0.269·20-s + 0.357·22-s − 0.338·23-s − 0.709·25-s − 0.665·26-s + 0.797·28-s − 1.02·29-s + 1.03·31-s + 0.176·32-s + 1.08·34-s + 0.859·35-s + 1.12·37-s + 0.0892·38-s + 0.190·40-s − 0.185·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.982341831\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.982341831\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 149 | \( 1 + T \) |
| good | 5 | \( 1 - 1.20T + 5T^{2} \) |
| 7 | \( 1 - 4.22T + 7T^{2} \) |
| 11 | \( 1 - 1.67T + 11T^{2} \) |
| 13 | \( 1 + 3.39T + 13T^{2} \) |
| 17 | \( 1 - 6.30T + 17T^{2} \) |
| 19 | \( 1 - 0.549T + 19T^{2} \) |
| 23 | \( 1 + 1.62T + 23T^{2} \) |
| 29 | \( 1 + 5.50T + 29T^{2} \) |
| 31 | \( 1 - 5.74T + 31T^{2} \) |
| 37 | \( 1 - 6.81T + 37T^{2} \) |
| 41 | \( 1 + 1.19T + 41T^{2} \) |
| 43 | \( 1 + 3.93T + 43T^{2} \) |
| 47 | \( 1 - 1.55T + 47T^{2} \) |
| 53 | \( 1 - 5.73T + 53T^{2} \) |
| 59 | \( 1 - 1.95T + 59T^{2} \) |
| 61 | \( 1 - 0.935T + 61T^{2} \) |
| 67 | \( 1 - 9.79T + 67T^{2} \) |
| 71 | \( 1 + 2.12T + 71T^{2} \) |
| 73 | \( 1 - 7.02T + 73T^{2} \) |
| 79 | \( 1 + 12.4T + 79T^{2} \) |
| 83 | \( 1 - 1.69T + 83T^{2} \) |
| 89 | \( 1 + 1.92T + 89T^{2} \) |
| 97 | \( 1 + 0.826T + 97T^{2} \) |
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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
−7.77797298763879867752216318159, −7.21585474508191928585983538238, −6.26135161163245884605596818773, −5.54024808220522433995950122268, −5.13166799788046624071469568641, −4.37080982183016967546260635558, −3.66596782547089409748075469799, −2.58015295384738946771843244753, −1.87813634381965607627039207557, −1.07788352999547408450728575655,
1.07788352999547408450728575655, 1.87813634381965607627039207557, 2.58015295384738946771843244753, 3.66596782547089409748075469799, 4.37080982183016967546260635558, 5.13166799788046624071469568641, 5.54024808220522433995950122268, 6.26135161163245884605596818773, 7.21585474508191928585983538238, 7.77797298763879867752216318159
