| L(s) = 1 | + 2-s + 4-s − 2·7-s + 8-s + 3.46·11-s − 5.46·13-s − 2·14-s + 16-s − 3.46·17-s + 19-s + 3.46·22-s + 6.92·23-s − 5.46·26-s − 2·28-s − 3.46·29-s − 1.46·31-s + 32-s − 3.46·34-s + 1.46·37-s + 38-s − 5.46·43-s + 3.46·44-s + 6.92·46-s + 6.92·47-s − 3·49-s − 5.46·52-s − 12.9·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.755·7-s + 0.353·8-s + 1.04·11-s − 1.51·13-s − 0.534·14-s + 0.250·16-s − 0.840·17-s + 0.229·19-s + 0.738·22-s + 1.44·23-s − 1.07·26-s − 0.377·28-s − 0.643·29-s − 0.262·31-s + 0.176·32-s − 0.594·34-s + 0.240·37-s + 0.162·38-s − 0.833·43-s + 0.522·44-s + 1.02·46-s + 1.01·47-s − 0.428·49-s − 0.757·52-s − 1.77·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 - 3.46T + 11T^{2} \) |
| 13 | \( 1 + 5.46T + 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 23 | \( 1 - 6.92T + 23T^{2} \) |
| 29 | \( 1 + 3.46T + 29T^{2} \) |
| 31 | \( 1 + 1.46T + 31T^{2} \) |
| 37 | \( 1 - 1.46T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 5.46T + 43T^{2} \) |
| 47 | \( 1 - 6.92T + 47T^{2} \) |
| 53 | \( 1 + 12.9T + 53T^{2} \) |
| 59 | \( 1 - 3.46T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 14.9T + 67T^{2} \) |
| 71 | \( 1 - 6.92T + 71T^{2} \) |
| 73 | \( 1 - 4.92T + 73T^{2} \) |
| 79 | \( 1 + 1.46T + 79T^{2} \) |
| 83 | \( 1 + 2.53T + 83T^{2} \) |
| 89 | \( 1 + 6.92T + 89T^{2} \) |
| 97 | \( 1 + 18.3T + 97T^{2} \) |
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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
−7.10919783261138359579092708112, −6.80580520404008461037141790375, −6.11006927922105614832268231763, −5.22960932950077418406640748047, −4.66338194636155862444136035613, −3.89564860960273519275571740348, −3.11210517974203165509137301249, −2.44406911474888411034033916247, −1.40511420006487770205224781131, 0,
1.40511420006487770205224781131, 2.44406911474888411034033916247, 3.11210517974203165509137301249, 3.89564860960273519275571740348, 4.66338194636155862444136035613, 5.22960932950077418406640748047, 6.11006927922105614832268231763, 6.80580520404008461037141790375, 7.10919783261138359579092708112
