| L(s) = 1 | + 2.79·2-s + 2.78·3-s + 5.82·4-s − 3.15·5-s + 7.79·6-s − 1.38·7-s + 10.6·8-s + 4.77·9-s − 8.81·10-s − 11-s + 16.2·12-s + 1.25·13-s − 3.86·14-s − 8.78·15-s + 18.2·16-s − 0.295·17-s + 13.3·18-s − 6.87·19-s − 18.3·20-s − 3.85·21-s − 2.79·22-s + 3.37·23-s + 29.8·24-s + 4.93·25-s + 3.50·26-s + 4.93·27-s − 8.05·28-s + ⋯ |
| L(s) = 1 | + 1.97·2-s + 1.60·3-s + 2.91·4-s − 1.40·5-s + 3.18·6-s − 0.522·7-s + 3.78·8-s + 1.59·9-s − 2.78·10-s − 0.301·11-s + 4.68·12-s + 0.347·13-s − 1.03·14-s − 2.26·15-s + 4.56·16-s − 0.0715·17-s + 3.14·18-s − 1.57·19-s − 4.10·20-s − 0.841·21-s − 0.596·22-s + 0.703·23-s + 6.08·24-s + 0.986·25-s + 0.687·26-s + 0.950·27-s − 1.52·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.627206187\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.627206187\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 131 | \( 1 - T \) |
| good | 2 | \( 1 - 2.79T + 2T^{2} \) |
| 3 | \( 1 - 2.78T + 3T^{2} \) |
| 5 | \( 1 + 3.15T + 5T^{2} \) |
| 7 | \( 1 + 1.38T + 7T^{2} \) |
| 13 | \( 1 - 1.25T + 13T^{2} \) |
| 17 | \( 1 + 0.295T + 17T^{2} \) |
| 19 | \( 1 + 6.87T + 19T^{2} \) |
| 23 | \( 1 - 3.37T + 23T^{2} \) |
| 29 | \( 1 + 1.71T + 29T^{2} \) |
| 31 | \( 1 + 2.46T + 31T^{2} \) |
| 37 | \( 1 - 10.2T + 37T^{2} \) |
| 41 | \( 1 + 6.08T + 41T^{2} \) |
| 43 | \( 1 + 7.92T + 43T^{2} \) |
| 47 | \( 1 + 2.75T + 47T^{2} \) |
| 53 | \( 1 + 6.96T + 53T^{2} \) |
| 59 | \( 1 + 2.02T + 59T^{2} \) |
| 61 | \( 1 + 9.27T + 61T^{2} \) |
| 67 | \( 1 + 0.0884T + 67T^{2} \) |
| 71 | \( 1 - 16.4T + 71T^{2} \) |
| 73 | \( 1 - 11.6T + 73T^{2} \) |
| 79 | \( 1 + 6.91T + 79T^{2} \) |
| 83 | \( 1 - 17.3T + 83T^{2} \) |
| 89 | \( 1 - 10.1T + 89T^{2} \) |
| 97 | \( 1 + 14.7T + 97T^{2} \) |
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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
−9.507401020242446487370330128675, −8.201870524006984089268326686867, −7.930290776892812521119300056655, −6.96205392907001904739848770393, −6.34346297039280907365356847807, −4.91195621339396347406361697610, −4.15641520403028570111000652602, −3.51816282358252241185645017686, −2.97540317459869303338976903939, −1.94885651767075860268243165367,
1.94885651767075860268243165367, 2.97540317459869303338976903939, 3.51816282358252241185645017686, 4.15641520403028570111000652602, 4.91195621339396347406361697610, 6.34346297039280907365356847807, 6.96205392907001904739848770393, 7.930290776892812521119300056655, 8.201870524006984089268326686867, 9.507401020242446487370330128675
