L(s) = 1 | − 2.56·2-s + 4.56·4-s + 0.438·7-s − 6.56·8-s − 1.56·11-s + 13-s − 1.12·14-s + 7.68·16-s − 1.56·17-s − 5.12·19-s + 4·22-s − 2.43·23-s − 2.56·26-s + 2·28-s − 7.12·29-s + 6·31-s − 6.56·32-s + 4·34-s + 10.6·37-s + 13.1·38-s + 3.56·41-s − 3.12·43-s − 7.12·44-s + 6.24·46-s − 11.1·47-s − 6.80·49-s + 4.56·52-s + ⋯ |
L(s) = 1 | − 1.81·2-s + 2.28·4-s + 0.165·7-s − 2.31·8-s − 0.470·11-s + 0.277·13-s − 0.300·14-s + 1.92·16-s − 0.378·17-s − 1.17·19-s + 0.852·22-s − 0.508·23-s − 0.502·26-s + 0.377·28-s − 1.32·29-s + 1.07·31-s − 1.15·32-s + 0.685·34-s + 1.75·37-s + 2.12·38-s + 0.556·41-s − 0.476·43-s − 1.07·44-s + 0.920·46-s − 1.62·47-s − 0.972·49-s + 0.632·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5915060072\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5915060072\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 2 | \( 1 + 2.56T + 2T^{2} \) |
| 7 | \( 1 - 0.438T + 7T^{2} \) |
| 11 | \( 1 + 1.56T + 11T^{2} \) |
| 17 | \( 1 + 1.56T + 17T^{2} \) |
| 19 | \( 1 + 5.12T + 19T^{2} \) |
| 23 | \( 1 + 2.43T + 23T^{2} \) |
| 29 | \( 1 + 7.12T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 - 10.6T + 37T^{2} \) |
| 41 | \( 1 - 3.56T + 41T^{2} \) |
| 43 | \( 1 + 3.12T + 43T^{2} \) |
| 47 | \( 1 + 11.1T + 47T^{2} \) |
| 53 | \( 1 - 4.68T + 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 + 6.68T + 61T^{2} \) |
| 67 | \( 1 - 11.3T + 67T^{2} \) |
| 71 | \( 1 - 10.4T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 - 4.68T + 79T^{2} \) |
| 83 | \( 1 - 16.4T + 83T^{2} \) |
| 89 | \( 1 - 10.6T + 89T^{2} \) |
| 97 | \( 1 + 16.9T + 97T^{2} \) |
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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
−8.669802789411637988215261537362, −8.102709722292768512409734764168, −7.65698270380819280294888523190, −6.58334641197981666927230891619, −6.21920489423314869167322112560, −4.99614723879208031226798095871, −3.82557996177240797007616237295, −2.55024213533629244986545216833, −1.87759195166261424484489187406, −0.60372834305678554809227235106,
0.60372834305678554809227235106, 1.87759195166261424484489187406, 2.55024213533629244986545216833, 3.82557996177240797007616237295, 4.99614723879208031226798095871, 6.21920489423314869167322112560, 6.58334641197981666927230891619, 7.65698270380819280294888523190, 8.102709722292768512409734764168, 8.669802789411637988215261537362