| L(s) = 1 | + 2-s − 2.44·3-s + 4-s − 2.44·6-s + 2.44·7-s + 8-s + 2.99·9-s − 11-s − 2.44·12-s + 5.89·13-s + 2.44·14-s + 16-s + 0.449·17-s + 2.99·18-s − 5.99·21-s − 22-s − 8.89·23-s − 2.44·24-s − 5·25-s + 5.89·26-s + 2.44·28-s − 7.89·29-s − 2·31-s + 32-s + 2.44·33-s + 0.449·34-s + 2.99·36-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.41·3-s + 0.5·4-s − 0.999·6-s + 0.925·7-s + 0.353·8-s + 0.999·9-s − 0.301·11-s − 0.707·12-s + 1.63·13-s + 0.654·14-s + 0.250·16-s + 0.109·17-s + 0.707·18-s − 1.30·21-s − 0.213·22-s − 1.85·23-s − 0.499·24-s − 25-s + 1.15·26-s + 0.462·28-s − 1.46·29-s − 0.359·31-s + 0.176·32-s + 0.426·33-s + 0.0770·34-s + 0.499·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 2.44T + 3T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 7 | \( 1 - 2.44T + 7T^{2} \) |
| 13 | \( 1 - 5.89T + 13T^{2} \) |
| 17 | \( 1 - 0.449T + 17T^{2} \) |
| 23 | \( 1 + 8.89T + 23T^{2} \) |
| 29 | \( 1 + 7.89T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 + 3.55T + 37T^{2} \) |
| 41 | \( 1 - 12.4T + 41T^{2} \) |
| 43 | \( 1 - 4.55T + 43T^{2} \) |
| 47 | \( 1 + 11.4T + 47T^{2} \) |
| 53 | \( 1 + 11.7T + 53T^{2} \) |
| 59 | \( 1 - 6.89T + 59T^{2} \) |
| 61 | \( 1 + 1.89T + 61T^{2} \) |
| 67 | \( 1 + 14.4T + 67T^{2} \) |
| 71 | \( 1 + 2.55T + 71T^{2} \) |
| 73 | \( 1 + 9.55T + 73T^{2} \) |
| 79 | \( 1 - 0.449T + 79T^{2} \) |
| 83 | \( 1 + 0.550T + 83T^{2} \) |
| 89 | \( 1 - 9T + 89T^{2} \) |
| 97 | \( 1 - 5.89T + 97T^{2} \) |
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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
−7.50715588974599285022472663448, −6.36653547717209694475441394831, −5.92437875091621018519739719427, −5.61927712181219217773908645735, −4.71709202453815893378688768247, −4.13161076140994040186604589488, −3.40924267850282776054907235229, −2.00459655860393729570317068255, −1.36583646101038601914599074160, 0,
1.36583646101038601914599074160, 2.00459655860393729570317068255, 3.40924267850282776054907235229, 4.13161076140994040186604589488, 4.71709202453815893378688768247, 5.61927712181219217773908645735, 5.92437875091621018519739719427, 6.36653547717209694475441394831, 7.50715588974599285022472663448
