| L(s) = 1 | + 0.445·2-s + 1.80·3-s − 0.801·4-s + 0.801·6-s − 0.445·7-s − 0.801·8-s + 2.24·9-s − 1.44·12-s + 0.445·13-s − 0.198·14-s + 0.445·16-s + 18-s + 19-s − 0.801·21-s − 1.80·23-s − 1.44·24-s + 25-s + 0.198·26-s + 2.24·27-s + 0.356·28-s + 1.80·31-s + 32-s − 1.80·36-s + 0.445·38-s + 0.801·39-s − 0.356·42-s + 2·43-s + ⋯ |
| L(s) = 1 | + 0.445·2-s + 1.80·3-s − 0.801·4-s + 0.801·6-s − 0.445·7-s − 0.801·8-s + 2.24·9-s − 1.44·12-s + 0.445·13-s − 0.198·14-s + 0.445·16-s + 18-s + 19-s − 0.801·21-s − 1.80·23-s − 1.44·24-s + 25-s + 0.198·26-s + 2.24·27-s + 0.356·28-s + 1.80·31-s + 32-s − 1.80·36-s + 0.445·38-s + 0.801·39-s − 0.356·42-s + 2·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3743 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3743 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.451214235\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.451214235\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 - T \) |
| 197 | \( 1 - T \) |
| good | 2 | \( 1 - 0.445T + T^{2} \) |
| 3 | \( 1 - 1.80T + T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 7 | \( 1 + 0.445T + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - 0.445T + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 23 | \( 1 + 1.80T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - 1.80T + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - 2T + T^{2} \) |
| 47 | \( 1 + 0.445T + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + 1.80T + T^{2} \) |
| 67 | \( 1 - 1.80T + T^{2} \) |
| 71 | \( 1 + 1.24T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + 1.24T + T^{2} \) |
| 83 | \( 1 + 1.80T + T^{2} \) |
| 89 | \( 1 + 2T + T^{2} \) |
| 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
−8.587117815022236338763736016613, −8.181450685250138411945550088775, −7.47508792383607793637915803727, −6.49730710549253353027232869359, −5.66027883861358838119162667619, −4.48431688631091847704729275516, −4.05090067214514345280547333048, −3.14545100217024242822747220986, −2.69748917247791082975408484586, −1.31413714452532707822957125831,
1.31413714452532707822957125831, 2.69748917247791082975408484586, 3.14545100217024242822747220986, 4.05090067214514345280547333048, 4.48431688631091847704729275516, 5.66027883861358838119162667619, 6.49730710549253353027232869359, 7.47508792383607793637915803727, 8.181450685250138411945550088775, 8.587117815022236338763736016613
