| L(s) = 1 | − 2·5-s − 7-s − 6·11-s + 4·13-s + 2·17-s + 4·19-s + 23-s − 25-s − 10·29-s + 8·31-s + 2·35-s + 8·37-s + 2·41-s + 6·43-s + 12·47-s + 49-s + 12·53-s + 12·55-s + 6·59-s + 6·61-s − 8·65-s − 2·67-s + 16·71-s + 2·73-s + 6·77-s − 4·83-s − 4·85-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.377·7-s − 1.80·11-s + 1.10·13-s + 0.485·17-s + 0.917·19-s + 0.208·23-s − 1/5·25-s − 1.85·29-s + 1.43·31-s + 0.338·35-s + 1.31·37-s + 0.312·41-s + 0.914·43-s + 1.75·47-s + 1/7·49-s + 1.64·53-s + 1.61·55-s + 0.781·59-s + 0.768·61-s − 0.992·65-s − 0.244·67-s + 1.89·71-s + 0.234·73-s + 0.683·77-s − 0.439·83-s − 0.433·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.037817868\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.037817868\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 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
−13.73001707151211, −13.32471639928526, −12.83675443394533, −12.51220261443481, −11.66360720107289, −11.49146952392207, −10.90729264411553, −10.39070484348837, −9.975513003633913, −9.328429928095985, −8.822471750881472, −8.118420219828157, −7.802144219981800, −7.449806749680437, −6.864785446877641, −5.969544095183611, −5.654480004503768, −5.186272654814315, −4.344378693364415, −3.836596916124996, −3.378183087901251, −2.640808845799126, −2.208810147615415, −0.9355305530669374, −0.5820822999488563,
0.5820822999488563, 0.9355305530669374, 2.208810147615415, 2.640808845799126, 3.378183087901251, 3.836596916124996, 4.344378693364415, 5.186272654814315, 5.654480004503768, 5.969544095183611, 6.864785446877641, 7.449806749680437, 7.802144219981800, 8.118420219828157, 8.822471750881472, 9.328429928095985, 9.975513003633913, 10.39070484348837, 10.90729264411553, 11.49146952392207, 11.66360720107289, 12.51220261443481, 12.83675443394533, 13.32471639928526, 13.73001707151211
