| L(s) = 1 | + 5-s − 7-s − 3·9-s − 2·13-s + 6·17-s + 4·19-s + 2·23-s + 25-s + 2·29-s + 10·31-s − 35-s + 2·37-s + 12·41-s − 3·45-s + 49-s − 6·53-s − 6·59-s + 3·63-s − 2·65-s − 2·67-s + 8·71-s + 10·73-s + 9·81-s + 4·83-s + 6·85-s − 6·89-s + 2·91-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.377·7-s − 9-s − 0.554·13-s + 1.45·17-s + 0.917·19-s + 0.417·23-s + 1/5·25-s + 0.371·29-s + 1.79·31-s − 0.169·35-s + 0.328·37-s + 1.87·41-s − 0.447·45-s + 1/7·49-s − 0.824·53-s − 0.781·59-s + 0.377·63-s − 0.248·65-s − 0.244·67-s + 0.949·71-s + 1.17·73-s + 81-s + 0.439·83-s + 0.650·85-s − 0.635·89-s + 0.209·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 67760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.802334627\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.802334627\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−14.16284858230821, −13.86403439257440, −13.20016124397133, −12.60403435407951, −12.13169418896500, −11.81075305089841, −11.12767484855284, −10.65021132751932, −10.01866984975635, −9.513622725661626, −9.335530024798784, −8.486162923129471, −7.961690476104209, −7.586903506998700, −6.861062642280467, −6.192055123313211, −5.872419662844801, −5.210851904657974, −4.798595277471885, −3.964907859545607, −3.049325894131544, −2.959346715525908, −2.200239089190553, −1.140118612466789, −0.6364091709266693,
0.6364091709266693, 1.140118612466789, 2.200239089190553, 2.959346715525908, 3.049325894131544, 3.964907859545607, 4.798595277471885, 5.210851904657974, 5.872419662844801, 6.192055123313211, 6.861062642280467, 7.586903506998700, 7.961690476104209, 8.486162923129471, 9.335530024798784, 9.513622725661626, 10.01866984975635, 10.65021132751932, 11.12767484855284, 11.81075305089841, 12.13169418896500, 12.60403435407951, 13.20016124397133, 13.86403439257440, 14.16284858230821
