| L(s) = 1 | + 0.906·2-s − 3.21·3-s − 1.17·4-s − 2.91·6-s − 2.59·7-s − 2.88·8-s + 7.35·9-s + 0.741·11-s + 3.78·12-s + 3.78·13-s − 2.35·14-s − 0.258·16-s + 3.16·17-s + 6.67·18-s − 19-s + 8.35·21-s + 0.672·22-s − 0.570·23-s + 9.27·24-s + 3.43·26-s − 14.0·27-s + 3.05·28-s + 6·29-s + 5.83·31-s + 5.52·32-s − 2.38·33-s + 2.87·34-s + ⋯ |
| L(s) = 1 | + 0.641·2-s − 1.85·3-s − 0.588·4-s − 1.19·6-s − 0.981·7-s − 1.01·8-s + 2.45·9-s + 0.223·11-s + 1.09·12-s + 1.05·13-s − 0.629·14-s − 0.0647·16-s + 0.768·17-s + 1.57·18-s − 0.229·19-s + 1.82·21-s + 0.143·22-s − 0.119·23-s + 1.89·24-s + 0.673·26-s − 2.69·27-s + 0.577·28-s + 1.11·29-s + 1.04·31-s + 0.977·32-s − 0.415·33-s + 0.492·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7406636871\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7406636871\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 - 0.906T + 2T^{2} \) |
| 3 | \( 1 + 3.21T + 3T^{2} \) |
| 7 | \( 1 + 2.59T + 7T^{2} \) |
| 11 | \( 1 - 0.741T + 11T^{2} \) |
| 13 | \( 1 - 3.78T + 13T^{2} \) |
| 17 | \( 1 - 3.16T + 17T^{2} \) |
| 23 | \( 1 + 0.570T + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 5.83T + 31T^{2} \) |
| 37 | \( 1 - 1.40T + 37T^{2} \) |
| 41 | \( 1 + 3.83T + 41T^{2} \) |
| 43 | \( 1 - 2.59T + 43T^{2} \) |
| 47 | \( 1 + 5.08T + 47T^{2} \) |
| 53 | \( 1 - 0.160T + 53T^{2} \) |
| 59 | \( 1 - 8.35T + 59T^{2} \) |
| 61 | \( 1 + 8.57T + 61T^{2} \) |
| 67 | \( 1 - 14.8T + 67T^{2} \) |
| 71 | \( 1 - 3.64T + 71T^{2} \) |
| 73 | \( 1 - 10.8T + 73T^{2} \) |
| 79 | \( 1 - 1.83T + 79T^{2} \) |
| 83 | \( 1 - 4.19T + 83T^{2} \) |
| 89 | \( 1 + 16.9T + 89T^{2} \) |
| 97 | \( 1 - 3.78T + 97T^{2} \) |
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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
−11.20020273511201676132153580325, −10.18772366459226803995714671017, −9.607038281480831581765178944945, −8.285944194876819456103630056013, −6.67752458474272111826444254441, −6.21645071576023168405639739515, −5.39421644333512201886752838619, −4.44968527760017289098763587493, −3.45307914098692482568328632010, −0.804271158418313148776300199802,
0.804271158418313148776300199802, 3.45307914098692482568328632010, 4.44968527760017289098763587493, 5.39421644333512201886752838619, 6.21645071576023168405639739515, 6.67752458474272111826444254441, 8.285944194876819456103630056013, 9.607038281480831581765178944945, 10.18772366459226803995714671017, 11.20020273511201676132153580325
