| L(s) = 1 | + 2.30·2-s − 2.74·3-s + 3.32·4-s − 6.32·6-s + 4.14·7-s + 3.05·8-s + 4.51·9-s + 3.63·11-s − 9.11·12-s − 4.90·13-s + 9.57·14-s + 0.401·16-s − 1.12·17-s + 10.4·18-s + 1.01·19-s − 11.3·21-s + 8.39·22-s + 5.59·23-s − 8.37·24-s − 11.3·26-s − 4.14·27-s + 13.7·28-s + 1.09·29-s + 7.84·31-s − 5.18·32-s − 9.96·33-s − 2.58·34-s + ⋯ |
| L(s) = 1 | + 1.63·2-s − 1.58·3-s + 1.66·4-s − 2.58·6-s + 1.56·7-s + 1.08·8-s + 1.50·9-s + 1.09·11-s − 2.62·12-s − 1.36·13-s + 2.55·14-s + 0.100·16-s − 0.272·17-s + 2.45·18-s + 0.233·19-s − 2.48·21-s + 1.78·22-s + 1.16·23-s − 1.70·24-s − 2.21·26-s − 0.797·27-s + 2.60·28-s + 0.202·29-s + 1.40·31-s − 0.916·32-s − 1.73·33-s − 0.444·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.988794217\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.988794217\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 47 | \( 1 + T \) |
| good | 2 | \( 1 - 2.30T + 2T^{2} \) |
| 3 | \( 1 + 2.74T + 3T^{2} \) |
| 7 | \( 1 - 4.14T + 7T^{2} \) |
| 11 | \( 1 - 3.63T + 11T^{2} \) |
| 13 | \( 1 + 4.90T + 13T^{2} \) |
| 17 | \( 1 + 1.12T + 17T^{2} \) |
| 19 | \( 1 - 1.01T + 19T^{2} \) |
| 23 | \( 1 - 5.59T + 23T^{2} \) |
| 29 | \( 1 - 1.09T + 29T^{2} \) |
| 31 | \( 1 - 7.84T + 31T^{2} \) |
| 37 | \( 1 - 6.04T + 37T^{2} \) |
| 41 | \( 1 - 10.0T + 41T^{2} \) |
| 43 | \( 1 + 3.65T + 43T^{2} \) |
| 53 | \( 1 - 8.70T + 53T^{2} \) |
| 59 | \( 1 + 14.2T + 59T^{2} \) |
| 61 | \( 1 - 9.09T + 61T^{2} \) |
| 67 | \( 1 + 12.1T + 67T^{2} \) |
| 71 | \( 1 - 4.30T + 71T^{2} \) |
| 73 | \( 1 + 15.3T + 73T^{2} \) |
| 79 | \( 1 - 9.20T + 79T^{2} \) |
| 83 | \( 1 + 8.44T + 83T^{2} \) |
| 89 | \( 1 - 1.62T + 89T^{2} \) |
| 97 | \( 1 + 18.1T + 97T^{2} \) |
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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
−10.21972403995635922898185891946, −9.027699279308315014587447789797, −7.63231569956733264680090219193, −6.88495052105835484673950794868, −6.11779105023481292729622649441, −5.30185804432245663935675060041, −4.65763432408372860757261620567, −4.30571806340781922847673080786, −2.62706897708296414827884387734, −1.23197449986251702908290967843,
1.23197449986251702908290967843, 2.62706897708296414827884387734, 4.30571806340781922847673080786, 4.65763432408372860757261620567, 5.30185804432245663935675060041, 6.11779105023481292729622649441, 6.88495052105835484673950794868, 7.63231569956733264680090219193, 9.027699279308315014587447789797, 10.21972403995635922898185891946
