| L(s) = 1 | − 3-s − 3.26·5-s − 7-s + 9-s + 1.38·11-s − 3.69·13-s + 3.26·15-s − 4.95·17-s − 4.43·19-s + 21-s + 23-s + 5.64·25-s − 27-s − 6.66·29-s − 6.95·31-s − 1.38·33-s + 3.26·35-s + 6.66·37-s + 3.69·39-s + 6.66·41-s − 1.35·43-s − 3.26·45-s − 11.6·47-s + 49-s + 4.95·51-s − 9.06·53-s − 4.50·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.45·5-s − 0.377·7-s + 0.333·9-s + 0.416·11-s − 1.02·13-s + 0.842·15-s − 1.20·17-s − 1.01·19-s + 0.218·21-s + 0.208·23-s + 1.12·25-s − 0.192·27-s − 1.23·29-s − 1.24·31-s − 0.240·33-s + 0.551·35-s + 1.09·37-s + 0.591·39-s + 1.04·41-s − 0.206·43-s − 0.486·45-s − 1.69·47-s + 0.142·49-s + 0.693·51-s − 1.24·53-s − 0.607·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1286719479\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1286719479\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + 3.26T + 5T^{2} \) |
| 11 | \( 1 - 1.38T + 11T^{2} \) |
| 13 | \( 1 + 3.69T + 13T^{2} \) |
| 17 | \( 1 + 4.95T + 17T^{2} \) |
| 19 | \( 1 + 4.43T + 19T^{2} \) |
| 29 | \( 1 + 6.66T + 29T^{2} \) |
| 31 | \( 1 + 6.95T + 31T^{2} \) |
| 37 | \( 1 - 6.66T + 37T^{2} \) |
| 41 | \( 1 - 6.66T + 41T^{2} \) |
| 43 | \( 1 + 1.35T + 43T^{2} \) |
| 47 | \( 1 + 11.6T + 47T^{2} \) |
| 53 | \( 1 + 9.06T + 53T^{2} \) |
| 59 | \( 1 - 4.78T + 59T^{2} \) |
| 61 | \( 1 + 7.92T + 61T^{2} \) |
| 67 | \( 1 + 11.3T + 67T^{2} \) |
| 71 | \( 1 + 12.4T + 71T^{2} \) |
| 73 | \( 1 - 5.90T + 73T^{2} \) |
| 79 | \( 1 + 1.38T + 79T^{2} \) |
| 83 | \( 1 + 17.4T + 83T^{2} \) |
| 89 | \( 1 + 2.11T + 89T^{2} \) |
| 97 | \( 1 + 0.474T + 97T^{2} \) |
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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
−7.61123267544090142448439128574, −7.29264675704442791149880104779, −6.53500615774141241511320513193, −5.89129647842586634212853860235, −4.81170452713457837776414088006, −4.35492865404284588760414426758, −3.72676501159648313249508079585, −2.78075713017077747452494213559, −1.70794587150615257709954379124, −0.17797704720568139974971665194,
0.17797704720568139974971665194, 1.70794587150615257709954379124, 2.78075713017077747452494213559, 3.72676501159648313249508079585, 4.35492865404284588760414426758, 4.81170452713457837776414088006, 5.89129647842586634212853860235, 6.53500615774141241511320513193, 7.29264675704442791149880104779, 7.61123267544090142448439128574
