| L(s) = 1 | + 2-s − 2.29·3-s + 4-s − 5-s − 2.29·6-s − 4.86·7-s + 8-s + 2.27·9-s − 10-s + 5.18·11-s − 2.29·12-s + 0.351·13-s − 4.86·14-s + 2.29·15-s + 16-s − 3.00·17-s + 2.27·18-s + 1.17·19-s − 20-s + 11.1·21-s + 5.18·22-s − 7.07·23-s − 2.29·24-s + 25-s + 0.351·26-s + 1.66·27-s − 4.86·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.32·3-s + 0.5·4-s − 0.447·5-s − 0.937·6-s − 1.83·7-s + 0.353·8-s + 0.757·9-s − 0.316·10-s + 1.56·11-s − 0.662·12-s + 0.0974·13-s − 1.30·14-s + 0.592·15-s + 0.250·16-s − 0.729·17-s + 0.535·18-s + 0.269·19-s − 0.223·20-s + 2.43·21-s + 1.10·22-s − 1.47·23-s − 0.468·24-s + 0.200·25-s + 0.0688·26-s + 0.321·27-s − 0.919·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 601 | \( 1 - T \) |
| good | 3 | \( 1 + 2.29T + 3T^{2} \) |
| 7 | \( 1 + 4.86T + 7T^{2} \) |
| 11 | \( 1 - 5.18T + 11T^{2} \) |
| 13 | \( 1 - 0.351T + 13T^{2} \) |
| 17 | \( 1 + 3.00T + 17T^{2} \) |
| 19 | \( 1 - 1.17T + 19T^{2} \) |
| 23 | \( 1 + 7.07T + 23T^{2} \) |
| 29 | \( 1 - 1.56T + 29T^{2} \) |
| 31 | \( 1 - 4.54T + 31T^{2} \) |
| 37 | \( 1 - 3.19T + 37T^{2} \) |
| 41 | \( 1 - 2.43T + 41T^{2} \) |
| 43 | \( 1 - 8.67T + 43T^{2} \) |
| 47 | \( 1 + 1.71T + 47T^{2} \) |
| 53 | \( 1 + 11.2T + 53T^{2} \) |
| 59 | \( 1 + 2.89T + 59T^{2} \) |
| 61 | \( 1 + 4.11T + 61T^{2} \) |
| 67 | \( 1 - 7.34T + 67T^{2} \) |
| 71 | \( 1 + 0.875T + 71T^{2} \) |
| 73 | \( 1 - 4.65T + 73T^{2} \) |
| 79 | \( 1 - 8.99T + 79T^{2} \) |
| 83 | \( 1 + 10.1T + 83T^{2} \) |
| 89 | \( 1 - 12.2T + 89T^{2} \) |
| 97 | \( 1 - 0.409T + 97T^{2} \) |
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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
−7.30488751826863141500754786271, −6.61560410039868924626751874591, −6.20984031774667109059484526308, −5.93019921453429331806412941534, −4.75202758405298065825270874874, −4.08727341665834872739028933891, −3.51563502122735042246891082022, −2.53330346854692118195066273108, −1.06926880630499540588423582818, 0,
1.06926880630499540588423582818, 2.53330346854692118195066273108, 3.51563502122735042246891082022, 4.08727341665834872739028933891, 4.75202758405298065825270874874, 5.93019921453429331806412941534, 6.20984031774667109059484526308, 6.61560410039868924626751874591, 7.30488751826863141500754786271
