| L(s) = 1 | − 1.85·2-s + 1.63·3-s + 2.44·4-s − 1.99·5-s − 3.03·6-s − 0.229·7-s − 2.67·8-s + 1.67·9-s + 3.69·10-s + 1.21·11-s + 3.99·12-s + 0.425·14-s − 3.26·15-s + 2.51·16-s − 3.10·18-s − 4.86·20-s − 0.375·21-s − 2.24·22-s − 4.37·24-s + 2.97·25-s + 1.10·27-s − 0.559·28-s + 1.79·29-s + 6.05·30-s − 1.99·32-s + 1.98·33-s + 0.457·35-s + ⋯ |
| L(s) = 1 | − 1.85·2-s + 1.63·3-s + 2.44·4-s − 1.99·5-s − 3.03·6-s − 0.229·7-s − 2.67·8-s + 1.67·9-s + 3.69·10-s + 1.21·11-s + 3.99·12-s + 0.425·14-s − 3.26·15-s + 2.51·16-s − 3.10·18-s − 4.86·20-s − 0.375·21-s − 2.24·22-s − 4.37·24-s + 2.97·25-s + 1.10·27-s − 0.559·28-s + 1.79·29-s + 6.05·30-s − 1.99·32-s + 1.98·33-s + 0.457·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6106896271\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6106896271\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 1151 | \( 1 - T \) |
| good | 2 | \( 1 + 1.85T + T^{2} \) |
| 3 | \( 1 - 1.63T + T^{2} \) |
| 5 | \( 1 + 1.99T + T^{2} \) |
| 7 | \( 1 + 0.229T + T^{2} \) |
| 11 | \( 1 - 1.21T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - 1.79T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - 0.380T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + 0.818T + T^{2} \) |
| 47 | \( 1 - 1.90T + T^{2} \) |
| 53 | \( 1 + 1.08T + T^{2} \) |
| 59 | \( 1 - 1.97T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - 0.955T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + 1.94T + T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.640814658847832590736329862791, −8.923550989826140273108751164228, −8.359409765773833479734008894553, −8.002588140019579011481090444614, −7.11283343456812072162237724307, −6.68456833960096509291039613386, −4.29445913690397553425145982666, −3.45754608610305916240631473896, −2.64050596426540880834873145080, −1.12855445454651058013647277599,
1.12855445454651058013647277599, 2.64050596426540880834873145080, 3.45754608610305916240631473896, 4.29445913690397553425145982666, 6.68456833960096509291039613386, 7.11283343456812072162237724307, 8.002588140019579011481090444614, 8.359409765773833479734008894553, 8.923550989826140273108751164228, 9.640814658847832590736329862791
