| L(s) = 1 | − 1.56·2-s + 3-s + 0.438·4-s − 1.56·6-s − 2.56·7-s + 2.43·8-s + 9-s + 1.43·11-s + 0.438·12-s + 7.12·13-s + 4·14-s − 4.68·16-s + 3.12·17-s − 1.56·18-s + 6·19-s − 2.56·21-s − 2.24·22-s + 7.68·23-s + 2.43·24-s − 11.1·26-s + 27-s − 1.12·28-s − 5.43·29-s + 7.12·31-s + 2.43·32-s + 1.43·33-s − 4.87·34-s + ⋯ |
| L(s) = 1 | − 1.10·2-s + 0.577·3-s + 0.219·4-s − 0.637·6-s − 0.968·7-s + 0.862·8-s + 0.333·9-s + 0.433·11-s + 0.126·12-s + 1.97·13-s + 1.06·14-s − 1.17·16-s + 0.757·17-s − 0.368·18-s + 1.37·19-s − 0.558·21-s − 0.478·22-s + 1.60·23-s + 0.497·24-s − 2.18·26-s + 0.192·27-s − 0.212·28-s − 1.00·29-s + 1.27·31-s + 0.431·32-s + 0.250·33-s − 0.836·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.366275578\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.366275578\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 47 | \( 1 + T \) |
| good | 2 | \( 1 + 1.56T + 2T^{2} \) |
| 7 | \( 1 + 2.56T + 7T^{2} \) |
| 11 | \( 1 - 1.43T + 11T^{2} \) |
| 13 | \( 1 - 7.12T + 13T^{2} \) |
| 17 | \( 1 - 3.12T + 17T^{2} \) |
| 19 | \( 1 - 6T + 19T^{2} \) |
| 23 | \( 1 - 7.68T + 23T^{2} \) |
| 29 | \( 1 + 5.43T + 29T^{2} \) |
| 31 | \( 1 - 7.12T + 31T^{2} \) |
| 37 | \( 1 + 3.43T + 37T^{2} \) |
| 41 | \( 1 + 1.12T + 41T^{2} \) |
| 43 | \( 1 + 11.1T + 43T^{2} \) |
| 53 | \( 1 + 4.24T + 53T^{2} \) |
| 59 | \( 1 + 1.12T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 3.12T + 67T^{2} \) |
| 71 | \( 1 + 5.12T + 71T^{2} \) |
| 73 | \( 1 + 0.876T + 73T^{2} \) |
| 79 | \( 1 + 5.43T + 79T^{2} \) |
| 83 | \( 1 - 1.12T + 83T^{2} \) |
| 89 | \( 1 - 8.24T + 89T^{2} \) |
| 97 | \( 1 + 11.9T + 97T^{2} \) |
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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
−8.619839010287527726123048282061, −8.125885562189757696159443134353, −7.21337777476355516420931328990, −6.65399523212878187514296852989, −5.72564681948181904840733328703, −4.69152801212127144907335384724, −3.46829556796645367684705152729, −3.24310939014855361679617182597, −1.58207226202124541382049686365, −0.882930258623306524822922614315,
0.882930258623306524822922614315, 1.58207226202124541382049686365, 3.24310939014855361679617182597, 3.46829556796645367684705152729, 4.69152801212127144907335384724, 5.72564681948181904840733328703, 6.65399523212878187514296852989, 7.21337777476355516420931328990, 8.125885562189757696159443134353, 8.619839010287527726123048282061
