| L(s) = 1 | + 0.0966·2-s − 1.99·4-s − 0.508·7-s − 0.385·8-s − 2.02·11-s − 3.21·13-s − 0.0491·14-s + 3.94·16-s − 2.75·17-s − 3.77·19-s − 0.196·22-s − 0.271·23-s − 0.310·26-s + 1.01·28-s − 2.04·29-s − 0.0781·31-s + 1.15·32-s − 0.266·34-s + 37-s − 0.364·38-s − 4.50·41-s − 3.01·43-s + 4.03·44-s − 0.0262·46-s + 6.78·47-s − 6.74·49-s + 6.39·52-s + ⋯ |
| L(s) = 1 | + 0.0683·2-s − 0.995·4-s − 0.192·7-s − 0.136·8-s − 0.611·11-s − 0.891·13-s − 0.0131·14-s + 0.986·16-s − 0.667·17-s − 0.865·19-s − 0.0417·22-s − 0.0565·23-s − 0.0608·26-s + 0.191·28-s − 0.380·29-s − 0.0140·31-s + 0.203·32-s − 0.0456·34-s + 0.164·37-s − 0.0591·38-s − 0.704·41-s − 0.459·43-s + 0.608·44-s − 0.00386·46-s + 0.989·47-s − 0.963·49-s + 0.886·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6076563334\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6076563334\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 2 | \( 1 - 0.0966T + 2T^{2} \) |
| 7 | \( 1 + 0.508T + 7T^{2} \) |
| 11 | \( 1 + 2.02T + 11T^{2} \) |
| 13 | \( 1 + 3.21T + 13T^{2} \) |
| 17 | \( 1 + 2.75T + 17T^{2} \) |
| 19 | \( 1 + 3.77T + 19T^{2} \) |
| 23 | \( 1 + 0.271T + 23T^{2} \) |
| 29 | \( 1 + 2.04T + 29T^{2} \) |
| 31 | \( 1 + 0.0781T + 31T^{2} \) |
| 41 | \( 1 + 4.50T + 41T^{2} \) |
| 43 | \( 1 + 3.01T + 43T^{2} \) |
| 47 | \( 1 - 6.78T + 47T^{2} \) |
| 53 | \( 1 - 3.94T + 53T^{2} \) |
| 59 | \( 1 + 13.4T + 59T^{2} \) |
| 61 | \( 1 - 5.04T + 61T^{2} \) |
| 67 | \( 1 + 3.02T + 67T^{2} \) |
| 71 | \( 1 + 13.0T + 71T^{2} \) |
| 73 | \( 1 + 6.01T + 73T^{2} \) |
| 79 | \( 1 + 1.14T + 79T^{2} \) |
| 83 | \( 1 - 6.92T + 83T^{2} \) |
| 89 | \( 1 + 3.68T + 89T^{2} \) |
| 97 | \( 1 - 3.05T + 97T^{2} \) |
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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
−7.79282046624070423252665199128, −7.24708047136715632469764898730, −6.32260669617492198692828473795, −5.67290697906270622449184746454, −4.82287542875936338614401694785, −4.47244653567338606830802206975, −3.55294751730822688386056552554, −2.72966840509265850681224836076, −1.79231973104743399111222132965, −0.36819842167243277329656792303,
0.36819842167243277329656792303, 1.79231973104743399111222132965, 2.72966840509265850681224836076, 3.55294751730822688386056552554, 4.47244653567338606830802206975, 4.82287542875936338614401694785, 5.67290697906270622449184746454, 6.32260669617492198692828473795, 7.24708047136715632469764898730, 7.79282046624070423252665199128
