| L(s) = 1 | − 1.73·2-s + 0.999·4-s − 2·7-s + 1.73·8-s + 3.46·11-s + 3.46·14-s − 5·16-s − 6.92·17-s − 2·19-s − 5.99·22-s + 6.92·23-s − 5·25-s − 1.99·28-s + 6.92·29-s − 2·31-s + 5.19·32-s + 11.9·34-s − 2·37-s + 3.46·38-s − 6.92·41-s + 8·43-s + 3.46·44-s − 11.9·46-s + 10.3·47-s − 3·49-s + 8.66·50-s − 3.46·56-s + ⋯ |
| L(s) = 1 | − 1.22·2-s + 0.499·4-s − 0.755·7-s + 0.612·8-s + 1.04·11-s + 0.925·14-s − 1.25·16-s − 1.68·17-s − 0.458·19-s − 1.27·22-s + 1.44·23-s − 25-s − 0.377·28-s + 1.28·29-s − 0.359·31-s + 0.918·32-s + 2.05·34-s − 0.328·37-s + 0.561·38-s − 1.08·41-s + 1.21·43-s + 0.522·44-s − 1.76·46-s + 1.51·47-s − 0.428·49-s + 1.22·50-s − 0.462·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 1.73T + 2T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 - 3.46T + 11T^{2} \) |
| 17 | \( 1 + 6.92T + 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 - 6.92T + 23T^{2} \) |
| 29 | \( 1 - 6.92T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + 6.92T + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 - 10.3T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 3.46T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 + 14T + 67T^{2} \) |
| 71 | \( 1 + 3.46T + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + 10.3T + 83T^{2} \) |
| 89 | \( 1 + 6.92T + 89T^{2} \) |
| 97 | \( 1 - 10T + 97T^{2} \) |
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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
−8.939932819757655947582294898696, −8.717394945817194771911214024179, −7.50122640406391094299576029293, −6.79963178349652217329348536959, −6.19228113557220520313158332858, −4.74362666343720561565526139526, −3.96158984561161604411309874545, −2.61946421446322524837121717635, −1.38935355750951163595138786834, 0,
1.38935355750951163595138786834, 2.61946421446322524837121717635, 3.96158984561161604411309874545, 4.74362666343720561565526139526, 6.19228113557220520313158332858, 6.79963178349652217329348536959, 7.50122640406391094299576029293, 8.717394945817194771911214024179, 8.939932819757655947582294898696
