| L(s) = 1 | + 1.67·2-s − 3-s + 0.806·4-s + 5-s − 1.67·6-s − 3.63·7-s − 1.99·8-s + 9-s + 1.67·10-s − 1.61·11-s − 0.806·12-s − 6.09·14-s − 15-s − 4.96·16-s − 3.28·17-s + 1.67·18-s + 7.11·19-s + 0.806·20-s + 3.63·21-s − 2.70·22-s + 5.73·23-s + 1.99·24-s + 25-s − 27-s − 2.93·28-s + 7.44·29-s − 1.67·30-s + ⋯ |
| L(s) = 1 | + 1.18·2-s − 0.577·3-s + 0.403·4-s + 0.447·5-s − 0.683·6-s − 1.37·7-s − 0.707·8-s + 0.333·9-s + 0.529·10-s − 0.486·11-s − 0.232·12-s − 1.62·14-s − 0.258·15-s − 1.24·16-s − 0.797·17-s + 0.394·18-s + 1.63·19-s + 0.180·20-s + 0.793·21-s − 0.575·22-s + 1.19·23-s + 0.408·24-s + 0.200·25-s − 0.192·27-s − 0.554·28-s + 1.38·29-s − 0.305·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.112896133\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.112896133\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 1.67T + 2T^{2} \) |
| 7 | \( 1 + 3.63T + 7T^{2} \) |
| 11 | \( 1 + 1.61T + 11T^{2} \) |
| 17 | \( 1 + 3.28T + 17T^{2} \) |
| 19 | \( 1 - 7.11T + 19T^{2} \) |
| 23 | \( 1 - 5.73T + 23T^{2} \) |
| 29 | \( 1 - 7.44T + 29T^{2} \) |
| 31 | \( 1 - 6.76T + 31T^{2} \) |
| 37 | \( 1 - 11.9T + 37T^{2} \) |
| 41 | \( 1 - 5.13T + 41T^{2} \) |
| 43 | \( 1 + 7.09T + 43T^{2} \) |
| 47 | \( 1 - 1.67T + 47T^{2} \) |
| 53 | \( 1 + 1.42T + 53T^{2} \) |
| 59 | \( 1 + 11.5T + 59T^{2} \) |
| 61 | \( 1 - 6.76T + 61T^{2} \) |
| 67 | \( 1 + 5.40T + 67T^{2} \) |
| 71 | \( 1 - 3.83T + 71T^{2} \) |
| 73 | \( 1 - 6.67T + 73T^{2} \) |
| 79 | \( 1 - 2.45T + 79T^{2} \) |
| 83 | \( 1 - 2.96T + 83T^{2} \) |
| 89 | \( 1 + 1.18T + 89T^{2} \) |
| 97 | \( 1 + 9.01T + 97T^{2} \) |
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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
−9.253587563320408554911162969874, −8.026223571083252632816267963014, −6.83286052920384996823232913556, −6.45865223067870368093597589525, −5.71616789534946700445191477142, −4.99040105157145281721464285649, −4.30086349831476222693897485273, −3.09702077200289714940615027602, −2.72778218947090044879568488531, −0.78061753795520610903443376000,
0.78061753795520610903443376000, 2.72778218947090044879568488531, 3.09702077200289714940615027602, 4.30086349831476222693897485273, 4.99040105157145281721464285649, 5.71616789534946700445191477142, 6.45865223067870368093597589525, 6.83286052920384996823232913556, 8.026223571083252632816267963014, 9.253587563320408554911162969874
