| L(s) = 1 | + 2.77·3-s − 3.10·5-s + 4.25·7-s + 4.69·9-s + 0.566·11-s + 6.64·13-s − 8.60·15-s + 0.193·17-s + 11.8·21-s − 1.91·23-s + 4.62·25-s + 4.68·27-s − 6.94·29-s + 6.20·31-s + 1.57·33-s − 13.2·35-s − 4.41·37-s + 18.4·39-s − 6.18·41-s + 2.00·43-s − 14.5·45-s + 11.4·47-s + 11.1·49-s + 0.536·51-s + 0.750·53-s − 1.75·55-s − 0.815·59-s + ⋯ |
| L(s) = 1 | + 1.60·3-s − 1.38·5-s + 1.60·7-s + 1.56·9-s + 0.170·11-s + 1.84·13-s − 2.22·15-s + 0.0469·17-s + 2.57·21-s − 0.399·23-s + 0.924·25-s + 0.901·27-s − 1.29·29-s + 1.11·31-s + 0.273·33-s − 2.23·35-s − 0.725·37-s + 2.94·39-s − 0.966·41-s + 0.306·43-s − 2.16·45-s + 1.67·47-s + 1.58·49-s + 0.0751·51-s + 0.103·53-s − 0.237·55-s − 0.106·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.455479791\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.455479791\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 2.77T + 3T^{2} \) |
| 5 | \( 1 + 3.10T + 5T^{2} \) |
| 7 | \( 1 - 4.25T + 7T^{2} \) |
| 11 | \( 1 - 0.566T + 11T^{2} \) |
| 13 | \( 1 - 6.64T + 13T^{2} \) |
| 17 | \( 1 - 0.193T + 17T^{2} \) |
| 23 | \( 1 + 1.91T + 23T^{2} \) |
| 29 | \( 1 + 6.94T + 29T^{2} \) |
| 31 | \( 1 - 6.20T + 31T^{2} \) |
| 37 | \( 1 + 4.41T + 37T^{2} \) |
| 41 | \( 1 + 6.18T + 41T^{2} \) |
| 43 | \( 1 - 2.00T + 43T^{2} \) |
| 47 | \( 1 - 11.4T + 47T^{2} \) |
| 53 | \( 1 - 0.750T + 53T^{2} \) |
| 59 | \( 1 + 0.815T + 59T^{2} \) |
| 61 | \( 1 - 0.738T + 61T^{2} \) |
| 67 | \( 1 - 7.41T + 67T^{2} \) |
| 71 | \( 1 + 4.98T + 71T^{2} \) |
| 73 | \( 1 + 7.58T + 73T^{2} \) |
| 79 | \( 1 - 11.5T + 79T^{2} \) |
| 83 | \( 1 + 2.71T + 83T^{2} \) |
| 89 | \( 1 - 3.28T + 89T^{2} \) |
| 97 | \( 1 + 13.0T + 97T^{2} \) |
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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
−8.723813730311350406114739304932, −7.904777728686675897238868081648, −7.83382480477897477228108401124, −6.85228967742868853990881513741, −5.59385544475121389864457913584, −4.42879823299282210686731380284, −3.91622901959441559208131231204, −3.32695835051530302361082071948, −2.09287322719689974222647382170, −1.19039526031485848625398454332,
1.19039526031485848625398454332, 2.09287322719689974222647382170, 3.32695835051530302361082071948, 3.91622901959441559208131231204, 4.42879823299282210686731380284, 5.59385544475121389864457913584, 6.85228967742868853990881513741, 7.83382480477897477228108401124, 7.904777728686675897238868081648, 8.723813730311350406114739304932
