| L(s) = 1 | + 0.339·2-s − 1.88·4-s − 0.383·5-s + 0.840·7-s − 1.31·8-s − 0.130·10-s − 11-s + 0.607·13-s + 0.285·14-s + 3.32·16-s + 1.38·17-s + 0.701·19-s + 0.722·20-s − 0.339·22-s − 2.92·23-s − 4.85·25-s + 0.206·26-s − 1.58·28-s + 5.31·29-s − 1.25·31-s + 3.76·32-s + 0.469·34-s − 0.322·35-s − 5.54·37-s + 0.238·38-s + 0.505·40-s + 3.91·41-s + ⋯ |
| L(s) = 1 | + 0.239·2-s − 0.942·4-s − 0.171·5-s + 0.317·7-s − 0.466·8-s − 0.0411·10-s − 0.301·11-s + 0.168·13-s + 0.0762·14-s + 0.830·16-s + 0.335·17-s + 0.160·19-s + 0.161·20-s − 0.0723·22-s − 0.609·23-s − 0.970·25-s + 0.0404·26-s − 0.299·28-s + 0.987·29-s − 0.226·31-s + 0.665·32-s + 0.0805·34-s − 0.0544·35-s − 0.911·37-s + 0.0386·38-s + 0.0799·40-s + 0.610·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 - 0.339T + 2T^{2} \) |
| 5 | \( 1 + 0.383T + 5T^{2} \) |
| 7 | \( 1 - 0.840T + 7T^{2} \) |
| 13 | \( 1 - 0.607T + 13T^{2} \) |
| 17 | \( 1 - 1.38T + 17T^{2} \) |
| 19 | \( 1 - 0.701T + 19T^{2} \) |
| 23 | \( 1 + 2.92T + 23T^{2} \) |
| 29 | \( 1 - 5.31T + 29T^{2} \) |
| 31 | \( 1 + 1.25T + 31T^{2} \) |
| 37 | \( 1 + 5.54T + 37T^{2} \) |
| 41 | \( 1 - 3.91T + 41T^{2} \) |
| 43 | \( 1 + 2.02T + 43T^{2} \) |
| 47 | \( 1 - 2.54T + 47T^{2} \) |
| 53 | \( 1 - 0.788T + 53T^{2} \) |
| 59 | \( 1 - 6.06T + 59T^{2} \) |
| 67 | \( 1 - 9.43T + 67T^{2} \) |
| 71 | \( 1 - 3.74T + 71T^{2} \) |
| 73 | \( 1 + 0.514T + 73T^{2} \) |
| 79 | \( 1 + 2.04T + 79T^{2} \) |
| 83 | \( 1 + 2.44T + 83T^{2} \) |
| 89 | \( 1 - 3.99T + 89T^{2} \) |
| 97 | \( 1 + 6.58T + 97T^{2} \) |
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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
−7.988790810114743977535966481022, −7.05167241287782160713171755242, −6.13800448112674886585289040262, −5.44609608302409043038966429485, −4.85378021585140895782006400102, −4.03044176978428214592491062489, −3.46271802999199478161783862306, −2.40375057441021109216992651210, −1.20676393172085402835458417823, 0,
1.20676393172085402835458417823, 2.40375057441021109216992651210, 3.46271802999199478161783862306, 4.03044176978428214592491062489, 4.85378021585140895782006400102, 5.44609608302409043038966429485, 6.13800448112674886585289040262, 7.05167241287782160713171755242, 7.988790810114743977535966481022
