| L(s) = 1 | − 1.33·2-s − 0.209·4-s + 1.27·7-s + 2.95·8-s − 1.16·11-s − 3.61·13-s − 1.71·14-s − 3.53·16-s − 5.35·17-s − 7.61·19-s + 1.55·22-s − 3.41·23-s + 4.83·26-s − 0.267·28-s + 3.21·29-s − 1.09·31-s − 1.17·32-s + 7.17·34-s − 7.80·37-s + 10.1·38-s + 3.00·41-s − 3.42·43-s + 0.243·44-s + 4.57·46-s + 9.41·47-s − 5.36·49-s + 0.754·52-s + ⋯ |
| L(s) = 1 | − 0.946·2-s − 0.104·4-s + 0.483·7-s + 1.04·8-s − 0.351·11-s − 1.00·13-s − 0.457·14-s − 0.884·16-s − 1.29·17-s − 1.74·19-s + 0.332·22-s − 0.712·23-s + 0.947·26-s − 0.0505·28-s + 0.597·29-s − 0.196·31-s − 0.208·32-s + 1.22·34-s − 1.28·37-s + 1.65·38-s + 0.469·41-s − 0.521·43-s + 0.0367·44-s + 0.674·46-s + 1.37·47-s − 0.766·49-s + 0.104·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5058187195\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5058187195\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 1.33T + 2T^{2} \) |
| 7 | \( 1 - 1.27T + 7T^{2} \) |
| 11 | \( 1 + 1.16T + 11T^{2} \) |
| 13 | \( 1 + 3.61T + 13T^{2} \) |
| 17 | \( 1 + 5.35T + 17T^{2} \) |
| 19 | \( 1 + 7.61T + 19T^{2} \) |
| 23 | \( 1 + 3.41T + 23T^{2} \) |
| 29 | \( 1 - 3.21T + 29T^{2} \) |
| 31 | \( 1 + 1.09T + 31T^{2} \) |
| 37 | \( 1 + 7.80T + 37T^{2} \) |
| 41 | \( 1 - 3.00T + 41T^{2} \) |
| 43 | \( 1 + 3.42T + 43T^{2} \) |
| 47 | \( 1 - 9.41T + 47T^{2} \) |
| 53 | \( 1 - 7.64T + 53T^{2} \) |
| 59 | \( 1 + 12.7T + 59T^{2} \) |
| 61 | \( 1 + 10.1T + 61T^{2} \) |
| 67 | \( 1 - 8.78T + 67T^{2} \) |
| 71 | \( 1 - 15.0T + 71T^{2} \) |
| 73 | \( 1 - 13.1T + 73T^{2} \) |
| 79 | \( 1 + 11.5T + 79T^{2} \) |
| 83 | \( 1 - 10.7T + 83T^{2} \) |
| 89 | \( 1 - 4.51T + 89T^{2} \) |
| 97 | \( 1 - 18.0T + 97T^{2} \) |
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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
−8.236854729977236131933485926001, −7.64876450793209664481437422842, −6.89667203852231722215409017654, −6.19531977085907456090967506198, −5.00405344907279420828765024836, −4.62042383526340905597474430842, −3.81288624515131638895155628402, −2.34829420733133660830467404184, −1.89299070221160507341514633484, −0.42313518584164741259639134421,
0.42313518584164741259639134421, 1.89299070221160507341514633484, 2.34829420733133660830467404184, 3.81288624515131638895155628402, 4.62042383526340905597474430842, 5.00405344907279420828765024836, 6.19531977085907456090967506198, 6.89667203852231722215409017654, 7.64876450793209664481437422842, 8.236854729977236131933485926001
