| L(s) = 1 | − 5-s + 2·11-s + 13-s + 17-s − 3·19-s − 4·23-s + 25-s − 2·29-s + 4·31-s + 41-s + 5·43-s − 6·47-s + 14·53-s − 2·55-s + 10·59-s + 4·61-s − 65-s + 2·67-s − 9·71-s − 7·73-s − 15·79-s − 9·83-s − 85-s − 11·89-s + 3·95-s + 5·97-s + 101-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.603·11-s + 0.277·13-s + 0.242·17-s − 0.688·19-s − 0.834·23-s + 1/5·25-s − 0.371·29-s + 0.718·31-s + 0.156·41-s + 0.762·43-s − 0.875·47-s + 1.92·53-s − 0.269·55-s + 1.30·59-s + 0.512·61-s − 0.124·65-s + 0.244·67-s − 1.06·71-s − 0.819·73-s − 1.68·79-s − 0.987·83-s − 0.108·85-s − 1.16·89-s + 0.307·95-s + 0.507·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.955003024\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.955003024\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 - T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + 15 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 11 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.01274797264463, −12.47421545652625, −11.87427092408072, −11.54334479652650, −11.30584582483724, −10.47600999698024, −10.18977085814587, −9.780300823371929, −9.075597598687737, −8.592467083376432, −8.396325769799614, −7.704576432502785, −7.206678144696579, −6.836765989533646, −6.120526227932581, −5.859765736765811, −5.224887280513856, −4.502346410503684, −4.091025705868595, −3.744010661742438, −3.013866754497342, −2.441122970956768, −1.784756840266688, −1.117049733544250, −0.4188598304594631,
0.4188598304594631, 1.117049733544250, 1.784756840266688, 2.441122970956768, 3.013866754497342, 3.744010661742438, 4.091025705868595, 4.502346410503684, 5.224887280513856, 5.859765736765811, 6.120526227932581, 6.836765989533646, 7.206678144696579, 7.704576432502785, 8.396325769799614, 8.592467083376432, 9.075597598687737, 9.780300823371929, 10.18977085814587, 10.47600999698024, 11.30584582483724, 11.54334479652650, 11.87427092408072, 12.47421545652625, 13.01274797264463
