| L(s) = 1 | + 1.23·2-s − 0.474·4-s − 1.76·5-s − 1.12·7-s − 3.05·8-s − 2.17·10-s + 5.66·11-s − 6.79·13-s − 1.39·14-s − 2.82·16-s + 0.225·17-s − 1.54·19-s + 0.836·20-s + 6.99·22-s + 1.66·23-s − 1.88·25-s − 8.39·26-s + 0.533·28-s + 1.14·29-s − 3.83·31-s + 2.61·32-s + 0.278·34-s + 1.98·35-s − 5.00·37-s − 1.90·38-s + 5.39·40-s + 9.29·41-s + ⋯ |
| L(s) = 1 | + 0.873·2-s − 0.237·4-s − 0.788·5-s − 0.425·7-s − 1.08·8-s − 0.689·10-s + 1.70·11-s − 1.88·13-s − 0.371·14-s − 0.706·16-s + 0.0546·17-s − 0.354·19-s + 0.186·20-s + 1.49·22-s + 0.347·23-s − 0.377·25-s − 1.64·26-s + 0.100·28-s + 0.212·29-s − 0.688·31-s + 0.463·32-s + 0.0477·34-s + 0.335·35-s − 0.823·37-s − 0.309·38-s + 0.852·40-s + 1.45·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6021 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6021 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.407188649\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.407188649\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 223 | \( 1 - T \) |
| good | 2 | \( 1 - 1.23T + 2T^{2} \) |
| 5 | \( 1 + 1.76T + 5T^{2} \) |
| 7 | \( 1 + 1.12T + 7T^{2} \) |
| 11 | \( 1 - 5.66T + 11T^{2} \) |
| 13 | \( 1 + 6.79T + 13T^{2} \) |
| 17 | \( 1 - 0.225T + 17T^{2} \) |
| 19 | \( 1 + 1.54T + 19T^{2} \) |
| 23 | \( 1 - 1.66T + 23T^{2} \) |
| 29 | \( 1 - 1.14T + 29T^{2} \) |
| 31 | \( 1 + 3.83T + 31T^{2} \) |
| 37 | \( 1 + 5.00T + 37T^{2} \) |
| 41 | \( 1 - 9.29T + 41T^{2} \) |
| 43 | \( 1 - 3.23T + 43T^{2} \) |
| 47 | \( 1 + 3.41T + 47T^{2} \) |
| 53 | \( 1 + 12.6T + 53T^{2} \) |
| 59 | \( 1 - 13.1T + 59T^{2} \) |
| 61 | \( 1 + 1.79T + 61T^{2} \) |
| 67 | \( 1 - 6.18T + 67T^{2} \) |
| 71 | \( 1 + 3.19T + 71T^{2} \) |
| 73 | \( 1 - 7.49T + 73T^{2} \) |
| 79 | \( 1 - 5.09T + 79T^{2} \) |
| 83 | \( 1 + 8.81T + 83T^{2} \) |
| 89 | \( 1 - 10.3T + 89T^{2} \) |
| 97 | \( 1 - 14.6T + 97T^{2} \) |
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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
−7.996996122215508861052181435582, −7.19700261340504860056262044726, −6.62601263272362420439482511349, −5.88698698944762190777649493194, −4.96789739529767217203013580361, −4.41716897827641255661441076741, −3.75701859544672338414446000444, −3.14568725985884232774529199110, −2.07058078364022069519863166552, −0.52749132210965435940991838734,
0.52749132210965435940991838734, 2.07058078364022069519863166552, 3.14568725985884232774529199110, 3.75701859544672338414446000444, 4.41716897827641255661441076741, 4.96789739529767217203013580361, 5.88698698944762190777649493194, 6.62601263272362420439482511349, 7.19700261340504860056262044726, 7.996996122215508861052181435582
