| L(s) = 1 | + 0.188·2-s − 3-s − 1.96·4-s − 1.33·5-s − 0.188·6-s − 0.912·7-s − 0.748·8-s + 9-s − 0.253·10-s − 11-s + 1.96·12-s + 5.78·13-s − 0.172·14-s + 1.33·15-s + 3.78·16-s + 3.08·17-s + 0.188·18-s − 6.38·19-s + 2.63·20-s + 0.912·21-s − 0.188·22-s + 7.04·23-s + 0.748·24-s − 3.20·25-s + 1.09·26-s − 27-s + 1.79·28-s + ⋯ |
| L(s) = 1 | + 0.133·2-s − 0.577·3-s − 0.982·4-s − 0.599·5-s − 0.0771·6-s − 0.345·7-s − 0.264·8-s + 0.333·9-s − 0.0800·10-s − 0.301·11-s + 0.567·12-s + 1.60·13-s − 0.0460·14-s + 0.345·15-s + 0.946·16-s + 0.749·17-s + 0.0445·18-s − 1.46·19-s + 0.588·20-s + 0.199·21-s − 0.0402·22-s + 1.46·23-s + 0.152·24-s − 0.640·25-s + 0.214·26-s − 0.192·27-s + 0.338·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 - 0.188T + 2T^{2} \) |
| 5 | \( 1 + 1.33T + 5T^{2} \) |
| 7 | \( 1 + 0.912T + 7T^{2} \) |
| 13 | \( 1 - 5.78T + 13T^{2} \) |
| 17 | \( 1 - 3.08T + 17T^{2} \) |
| 19 | \( 1 + 6.38T + 19T^{2} \) |
| 23 | \( 1 - 7.04T + 23T^{2} \) |
| 29 | \( 1 + 9.45T + 29T^{2} \) |
| 31 | \( 1 - 7.73T + 31T^{2} \) |
| 37 | \( 1 + 8.78T + 37T^{2} \) |
| 41 | \( 1 - 8.00T + 41T^{2} \) |
| 43 | \( 1 - 10.8T + 43T^{2} \) |
| 47 | \( 1 - 10.0T + 47T^{2} \) |
| 53 | \( 1 + 3.34T + 53T^{2} \) |
| 59 | \( 1 + 11.0T + 59T^{2} \) |
| 67 | \( 1 + 14.0T + 67T^{2} \) |
| 71 | \( 1 + 15.6T + 71T^{2} \) |
| 73 | \( 1 - 2.96T + 73T^{2} \) |
| 79 | \( 1 + 16.1T + 79T^{2} \) |
| 83 | \( 1 + 8.28T + 83T^{2} \) |
| 89 | \( 1 - 16.6T + 89T^{2} \) |
| 97 | \( 1 - 2.95T + 97T^{2} \) |
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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
−8.858105794266803138448644921915, −8.022350442101465660002519243051, −7.26855978670221330152862891131, −6.06189407340562648863251874165, −5.71703053543832418000812239900, −4.52449246658220004220409158478, −3.97201651872220731443097426683, −3.09059670178175732350319637948, −1.26022053077776529926831056568, 0,
1.26022053077776529926831056568, 3.09059670178175732350319637948, 3.97201651872220731443097426683, 4.52449246658220004220409158478, 5.71703053543832418000812239900, 6.06189407340562648863251874165, 7.26855978670221330152862891131, 8.022350442101465660002519243051, 8.858105794266803138448644921915
