| L(s) = 1 | − 2-s + 1.44·3-s + 4-s − 1.44·6-s − 2.44·7-s − 8-s − 0.898·9-s + 3.44·11-s + 1.44·12-s + 0.449·13-s + 2.44·14-s + 16-s − 3.44·17-s + 0.898·18-s − 5·19-s − 3.55·21-s − 3.44·22-s − 2·23-s − 1.44·24-s − 0.449·26-s − 5.65·27-s − 2.44·28-s − 0.898·29-s + 4.44·31-s − 32-s + 5·33-s + 3.44·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.836·3-s + 0.5·4-s − 0.591·6-s − 0.925·7-s − 0.353·8-s − 0.299·9-s + 1.04·11-s + 0.418·12-s + 0.124·13-s + 0.654·14-s + 0.250·16-s − 0.836·17-s + 0.211·18-s − 1.14·19-s − 0.774·21-s − 0.735·22-s − 0.417·23-s − 0.295·24-s − 0.0881·26-s − 1.08·27-s − 0.462·28-s − 0.166·29-s + 0.799·31-s − 0.176·32-s + 0.870·33-s + 0.591·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 3 | \( 1 - 1.44T + 3T^{2} \) |
| 7 | \( 1 + 2.44T + 7T^{2} \) |
| 11 | \( 1 - 3.44T + 11T^{2} \) |
| 13 | \( 1 - 0.449T + 13T^{2} \) |
| 17 | \( 1 + 3.44T + 17T^{2} \) |
| 19 | \( 1 + 5T + 19T^{2} \) |
| 23 | \( 1 + 2T + 23T^{2} \) |
| 29 | \( 1 + 0.898T + 29T^{2} \) |
| 31 | \( 1 - 4.44T + 31T^{2} \) |
| 41 | \( 1 - T + 41T^{2} \) |
| 43 | \( 1 - 1.10T + 43T^{2} \) |
| 47 | \( 1 + 9.79T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 2T + 59T^{2} \) |
| 61 | \( 1 + 6.44T + 61T^{2} \) |
| 67 | \( 1 + 4.55T + 67T^{2} \) |
| 71 | \( 1 + 7.55T + 71T^{2} \) |
| 73 | \( 1 + 12.7T + 73T^{2} \) |
| 79 | \( 1 - 7.79T + 79T^{2} \) |
| 83 | \( 1 + 3.44T + 83T^{2} \) |
| 89 | \( 1 - 14.3T + 89T^{2} \) |
| 97 | \( 1 + 14T + 97T^{2} \) |
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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
−8.864720741768348609016183474117, −8.334834854918181383278511702911, −7.42662619777200227219034265936, −6.39901597490605843398317361077, −6.15592620182016360245135055390, −4.51399061144460739509022220272, −3.56290718360916960598390387838, −2.73885823713996759335050505992, −1.72506485647856065953250782080, 0,
1.72506485647856065953250782080, 2.73885823713996759335050505992, 3.56290718360916960598390387838, 4.51399061144460739509022220272, 6.15592620182016360245135055390, 6.39901597490605843398317361077, 7.42662619777200227219034265936, 8.334834854918181383278511702911, 8.864720741768348609016183474117
