| L(s) = 1 | − 2.73·3-s + 5-s + 2.73·7-s + 4.46·9-s − 4.73·11-s − 4·13-s − 2.73·15-s − 17-s + 1.46·19-s − 7.46·21-s + 8.19·23-s + 25-s − 3.99·27-s − 3.46·29-s − 3.26·31-s + 12.9·33-s + 2.73·35-s − 0.535·37-s + 10.9·39-s − 3.46·41-s + 0.535·43-s + 4.46·45-s − 12.9·47-s + 0.464·49-s + 2.73·51-s + 6·53-s − 4.73·55-s + ⋯ |
| L(s) = 1 | − 1.57·3-s + 0.447·5-s + 1.03·7-s + 1.48·9-s − 1.42·11-s − 1.10·13-s − 0.705·15-s − 0.242·17-s + 0.335·19-s − 1.62·21-s + 1.70·23-s + 0.200·25-s − 0.769·27-s − 0.643·29-s − 0.586·31-s + 2.25·33-s + 0.461·35-s − 0.0881·37-s + 1.74·39-s − 0.541·41-s + 0.0817·43-s + 0.665·45-s − 1.88·47-s + 0.0663·49-s + 0.382·51-s + 0.824·53-s − 0.638·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 + 2.73T + 3T^{2} \) |
| 7 | \( 1 - 2.73T + 7T^{2} \) |
| 11 | \( 1 + 4.73T + 11T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 19 | \( 1 - 1.46T + 19T^{2} \) |
| 23 | \( 1 - 8.19T + 23T^{2} \) |
| 29 | \( 1 + 3.46T + 29T^{2} \) |
| 31 | \( 1 + 3.26T + 31T^{2} \) |
| 37 | \( 1 + 0.535T + 37T^{2} \) |
| 41 | \( 1 + 3.46T + 41T^{2} \) |
| 43 | \( 1 - 0.535T + 43T^{2} \) |
| 47 | \( 1 + 12.9T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + 2.53T + 59T^{2} \) |
| 61 | \( 1 + 4.92T + 61T^{2} \) |
| 67 | \( 1 - 10T + 67T^{2} \) |
| 71 | \( 1 + 11.6T + 71T^{2} \) |
| 73 | \( 1 - 6.39T + 73T^{2} \) |
| 79 | \( 1 + 14.5T + 79T^{2} \) |
| 83 | \( 1 + 8.53T + 83T^{2} \) |
| 89 | \( 1 - 4.39T + 89T^{2} \) |
| 97 | \( 1 + 4.92T + 97T^{2} \) |
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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
−9.426737054609399808742414894577, −8.251330306077523802405396945697, −7.36926926223187686397752337944, −6.73786045678305662885440027409, −5.42916087698959198947136827595, −5.26342107414819848000697702043, −4.55546737882726099236283732875, −2.79362646579380484496800247011, −1.52216316616934081123372279623, 0,
1.52216316616934081123372279623, 2.79362646579380484496800247011, 4.55546737882726099236283732875, 5.26342107414819848000697702043, 5.42916087698959198947136827595, 6.73786045678305662885440027409, 7.36926926223187686397752337944, 8.251330306077523802405396945697, 9.426737054609399808742414894577
