| L(s) = 1 | − 3.27·3-s − 2·4-s − 1.87·5-s − 2.27·7-s + 7.75·9-s + 2.40·11-s + 6.55·12-s + 13-s + 6.15·15-s + 4·16-s − 3.75·17-s − 0.402·19-s + 3.75·20-s + 7.47·21-s − 23-s − 1.47·25-s − 15.5·27-s + 4.55·28-s + 4.80·29-s − 4·31-s − 7.87·33-s + 4.27·35-s − 15.5·36-s + 10.0·37-s − 3.27·39-s + 12.3·41-s + 8.31·43-s + ⋯ |
| L(s) = 1 | − 1.89·3-s − 4-s − 0.839·5-s − 0.861·7-s + 2.58·9-s + 0.724·11-s + 1.89·12-s + 0.277·13-s + 1.58·15-s + 16-s − 0.910·17-s − 0.0922·19-s + 0.839·20-s + 1.63·21-s − 0.208·23-s − 0.295·25-s − 3.00·27-s + 0.861·28-s + 0.892·29-s − 0.718·31-s − 1.37·33-s + 0.723·35-s − 2.58·36-s + 1.64·37-s − 0.525·39-s + 1.92·41-s + 1.26·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 299 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 299 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3420768423\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3420768423\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 + 2T^{2} \) |
| 3 | \( 1 + 3.27T + 3T^{2} \) |
| 5 | \( 1 + 1.87T + 5T^{2} \) |
| 7 | \( 1 + 2.27T + 7T^{2} \) |
| 11 | \( 1 - 2.40T + 11T^{2} \) |
| 17 | \( 1 + 3.75T + 17T^{2} \) |
| 19 | \( 1 + 0.402T + 19T^{2} \) |
| 29 | \( 1 - 4.80T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 10.0T + 37T^{2} \) |
| 41 | \( 1 - 12.3T + 41T^{2} \) |
| 43 | \( 1 - 8.31T + 43T^{2} \) |
| 47 | \( 1 + 6.31T + 47T^{2} \) |
| 53 | \( 1 + 2.24T + 53T^{2} \) |
| 59 | \( 1 + 2.24T + 59T^{2} \) |
| 61 | \( 1 - 13.1T + 61T^{2} \) |
| 67 | \( 1 - 1.31T + 67T^{2} \) |
| 71 | \( 1 + 3.44T + 71T^{2} \) |
| 73 | \( 1 - 6.80T + 73T^{2} \) |
| 79 | \( 1 - 3.19T + 79T^{2} \) |
| 83 | \( 1 + 5.48T + 83T^{2} \) |
| 89 | \( 1 + 12.9T + 89T^{2} \) |
| 97 | \( 1 - 16.1T + 97T^{2} \) |
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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
−11.65843732240548903016029878857, −11.04426396892557142131587451799, −9.966967810552485218459729921171, −9.170376002383120129559089449108, −7.72983263408473013766462279047, −6.56071704416060001570343219938, −5.84152361905065030560150080052, −4.54154178697790193489059136233, −3.92900348001709415455822627401, −0.65592638904476860194450488482,
0.65592638904476860194450488482, 3.92900348001709415455822627401, 4.54154178697790193489059136233, 5.84152361905065030560150080052, 6.56071704416060001570343219938, 7.72983263408473013766462279047, 9.170376002383120129559089449108, 9.966967810552485218459729921171, 11.04426396892557142131587451799, 11.65843732240548903016029878857
