L(s) = 1 | + 5-s + 4·7-s − 3·9-s + 4·11-s + 2·17-s + 4·19-s + 4·23-s + 25-s + 2·29-s + 8·31-s + 4·35-s + 6·37-s + 6·41-s + 8·43-s − 3·45-s − 4·47-s + 9·49-s − 6·53-s + 4·55-s − 4·59-s + 2·61-s − 12·63-s + 8·67-s + 6·73-s + 16·77-s + 9·81-s − 16·83-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 1.51·7-s − 9-s + 1.20·11-s + 0.485·17-s + 0.917·19-s + 0.834·23-s + 1/5·25-s + 0.371·29-s + 1.43·31-s + 0.676·35-s + 0.986·37-s + 0.937·41-s + 1.21·43-s − 0.447·45-s − 0.583·47-s + 9/7·49-s − 0.824·53-s + 0.539·55-s − 0.520·59-s + 0.256·61-s − 1.51·63-s + 0.977·67-s + 0.702·73-s + 1.82·77-s + 81-s − 1.75·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.657875795\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.657875795\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−14.42348078809323, −14.04548033972279, −13.72076622065296, −12.82768086838973, −12.34124139020279, −11.69324890728209, −11.27765787259393, −11.20640274989453, −10.29260906753295, −9.770240192133980, −9.051151164743510, −8.887744336334850, −8.003160391213557, −7.856564914211415, −7.072370837446894, −6.334477625705576, −5.936247019670364, −5.264687534250796, −4.775611626839977, −4.248721042309023, −3.387502791587691, −2.757864233261148, −2.107630389840526, −1.158413653218510, −0.9201653962135482,
0.9201653962135482, 1.158413653218510, 2.107630389840526, 2.757864233261148, 3.387502791587691, 4.248721042309023, 4.775611626839977, 5.264687534250796, 5.936247019670364, 6.334477625705576, 7.072370837446894, 7.856564914211415, 8.003160391213557, 8.887744336334850, 9.051151164743510, 9.770240192133980, 10.29260906753295, 11.20640274989453, 11.27765787259393, 11.69324890728209, 12.34124139020279, 12.82768086838973, 13.72076622065296, 14.04548033972279, 14.42348078809323