L(s) = 1 | + 3-s + 5-s − 7-s + 9-s + 2·13-s + 15-s + 6·17-s − 4·19-s − 21-s − 8·23-s + 25-s + 27-s − 6·29-s − 8·31-s − 35-s + 6·37-s + 2·39-s + 6·41-s − 4·43-s + 45-s + 8·47-s + 49-s + 6·51-s − 10·53-s − 4·57-s + 12·59-s + 10·61-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s + 0.554·13-s + 0.258·15-s + 1.45·17-s − 0.917·19-s − 0.218·21-s − 1.66·23-s + 1/5·25-s + 0.192·27-s − 1.11·29-s − 1.43·31-s − 0.169·35-s + 0.986·37-s + 0.320·39-s + 0.937·41-s − 0.609·43-s + 0.149·45-s + 1.16·47-s + 1/7·49-s + 0.840·51-s − 1.37·53-s − 0.529·57-s + 1.56·59-s + 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 203280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 203280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−13.16201544839206, −12.95316984928526, −12.38634562228824, −12.08898926138185, −11.26871477115411, −10.93221180391242, −10.43197600921505, −9.799675767123544, −9.550047323808619, −9.218213200199860, −8.394462784457065, −8.124200245883305, −7.687298962224433, −7.073480983155974, −6.504351461264764, −6.051330845875993, −5.485390388463507, −5.215722676573007, −4.137844622821449, −3.811597867943401, −3.545998057709955, −2.560775337470023, −2.262669875405863, −1.586269800088367, −0.9134822963036496, 0,
0.9134822963036496, 1.586269800088367, 2.262669875405863, 2.560775337470023, 3.545998057709955, 3.811597867943401, 4.137844622821449, 5.215722676573007, 5.485390388463507, 6.051330845875993, 6.504351461264764, 7.073480983155974, 7.687298962224433, 8.124200245883305, 8.394462784457065, 9.218213200199860, 9.550047323808619, 9.799675767123544, 10.43197600921505, 10.93221180391242, 11.26871477115411, 12.08898926138185, 12.38634562228824, 12.95316984928526, 13.16201544839206