| L(s) = 1 | + 3-s + 5-s + 4.86·7-s + 9-s + 4.77·11-s + 2.77·13-s + 15-s − 0.636·17-s − 3.50·19-s + 4.86·21-s − 23-s + 25-s + 27-s + 7.36·29-s − 5.15·31-s + 4.77·33-s + 4.86·35-s + 2.86·37-s + 2.77·39-s − 6.19·41-s − 8.28·43-s + 45-s + 7.50·47-s + 16.6·49-s − 0.636·51-s − 4.91·53-s + 4.77·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1.83·7-s + 0.333·9-s + 1.44·11-s + 0.770·13-s + 0.258·15-s − 0.154·17-s − 0.804·19-s + 1.06·21-s − 0.208·23-s + 0.200·25-s + 0.192·27-s + 1.36·29-s − 0.925·31-s + 0.831·33-s + 0.822·35-s + 0.471·37-s + 0.444·39-s − 0.967·41-s − 1.26·43-s + 0.149·45-s + 1.09·47-s + 2.38·49-s − 0.0891·51-s − 0.675·53-s + 0.644·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.493382858\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.493382858\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 - 4.86T + 7T^{2} \) |
| 11 | \( 1 - 4.77T + 11T^{2} \) |
| 13 | \( 1 - 2.77T + 13T^{2} \) |
| 17 | \( 1 + 0.636T + 17T^{2} \) |
| 19 | \( 1 + 3.50T + 19T^{2} \) |
| 29 | \( 1 - 7.36T + 29T^{2} \) |
| 31 | \( 1 + 5.15T + 31T^{2} \) |
| 37 | \( 1 - 2.86T + 37T^{2} \) |
| 41 | \( 1 + 6.19T + 41T^{2} \) |
| 43 | \( 1 + 8.28T + 43T^{2} \) |
| 47 | \( 1 - 7.50T + 47T^{2} \) |
| 53 | \( 1 + 4.91T + 53T^{2} \) |
| 59 | \( 1 + 8.65T + 59T^{2} \) |
| 61 | \( 1 + 15.0T + 61T^{2} \) |
| 67 | \( 1 + 13.4T + 67T^{2} \) |
| 71 | \( 1 + 11.0T + 71T^{2} \) |
| 73 | \( 1 + 15.2T + 73T^{2} \) |
| 79 | \( 1 + 1.45T + 79T^{2} \) |
| 83 | \( 1 - 6.63T + 83T^{2} \) |
| 89 | \( 1 - 9.29T + 89T^{2} \) |
| 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
−8.775879532410289792729021515006, −8.239737333932905660467805424765, −7.42477086743039903609847073701, −6.52482091861030953420732166353, −5.80379878150984252555425488127, −4.63068674124301882825449585539, −4.26150526731732992308421867233, −3.10120492154221778662080669951, −1.78025880477834165508673640414, −1.41096349832736927798572668329,
1.41096349832736927798572668329, 1.78025880477834165508673640414, 3.10120492154221778662080669951, 4.26150526731732992308421867233, 4.63068674124301882825449585539, 5.80379878150984252555425488127, 6.52482091861030953420732166353, 7.42477086743039903609847073701, 8.239737333932905660467805424765, 8.775879532410289792729021515006
