| L(s) = 1 | + 3.03·3-s − 0.0964·5-s + 5.16·7-s + 6.20·9-s − 2.15·11-s − 3.03·13-s − 0.292·15-s + 3.71·17-s − 6.01·19-s + 15.6·21-s − 2.42·23-s − 4.99·25-s + 9.71·27-s + 5.72·29-s + 5.57·31-s − 6.53·33-s − 0.498·35-s + 3.42·37-s − 9.20·39-s + 10.0·41-s + 9.75·43-s − 0.598·45-s − 2.69·47-s + 19.6·49-s + 11.2·51-s − 2.96·53-s + 0.207·55-s + ⋯ |
| L(s) = 1 | + 1.75·3-s − 0.0431·5-s + 1.95·7-s + 2.06·9-s − 0.649·11-s − 0.841·13-s − 0.0755·15-s + 0.900·17-s − 1.38·19-s + 3.41·21-s − 0.506·23-s − 0.998·25-s + 1.86·27-s + 1.06·29-s + 1.00·31-s − 1.13·33-s − 0.0841·35-s + 0.563·37-s − 1.47·39-s + 1.56·41-s + 1.48·43-s − 0.0891·45-s − 0.393·47-s + 2.81·49-s + 1.57·51-s − 0.407·53-s + 0.0280·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.526334078\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.526334078\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 251 | \( 1 - T \) |
| good | 3 | \( 1 - 3.03T + 3T^{2} \) |
| 5 | \( 1 + 0.0964T + 5T^{2} \) |
| 7 | \( 1 - 5.16T + 7T^{2} \) |
| 11 | \( 1 + 2.15T + 11T^{2} \) |
| 13 | \( 1 + 3.03T + 13T^{2} \) |
| 17 | \( 1 - 3.71T + 17T^{2} \) |
| 19 | \( 1 + 6.01T + 19T^{2} \) |
| 23 | \( 1 + 2.42T + 23T^{2} \) |
| 29 | \( 1 - 5.72T + 29T^{2} \) |
| 31 | \( 1 - 5.57T + 31T^{2} \) |
| 37 | \( 1 - 3.42T + 37T^{2} \) |
| 41 | \( 1 - 10.0T + 41T^{2} \) |
| 43 | \( 1 - 9.75T + 43T^{2} \) |
| 47 | \( 1 + 2.69T + 47T^{2} \) |
| 53 | \( 1 + 2.96T + 53T^{2} \) |
| 59 | \( 1 + 1.35T + 59T^{2} \) |
| 61 | \( 1 + 9.81T + 61T^{2} \) |
| 67 | \( 1 - 8.93T + 67T^{2} \) |
| 71 | \( 1 + 3.08T + 71T^{2} \) |
| 73 | \( 1 - 2.54T + 73T^{2} \) |
| 79 | \( 1 - 17.3T + 79T^{2} \) |
| 83 | \( 1 - 5.00T + 83T^{2} \) |
| 89 | \( 1 - 11.2T + 89T^{2} \) |
| 97 | \( 1 - 0.180T + 97T^{2} \) |
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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
−8.176531655526994949919704930232, −7.82364158964356613414736064846, −7.64331520758047637092721544881, −6.34135126667890080353124960314, −5.19813861913859172639777660428, −4.46057694746688938681815968988, −3.94733193253577594990675548813, −2.55886815877821666420943192617, −2.31637412160956543225573925163, −1.24567800710590617894429908184,
1.24567800710590617894429908184, 2.31637412160956543225573925163, 2.55886815877821666420943192617, 3.94733193253577594990675548813, 4.46057694746688938681815968988, 5.19813861913859172639777660428, 6.34135126667890080353124960314, 7.64331520758047637092721544881, 7.82364158964356613414736064846, 8.176531655526994949919704930232
