| L(s) = 1 | + 2·2-s + 3-s + 2·4-s + 5-s + 2·6-s − 3·7-s + 9-s + 2·10-s − 5·11-s + 2·12-s + 13-s − 6·14-s + 15-s − 4·16-s + 5·17-s + 2·18-s + 2·19-s + 2·20-s − 3·21-s − 10·22-s − 23-s + 25-s + 2·26-s + 27-s − 6·28-s + 10·29-s + 2·30-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s + 4-s + 0.447·5-s + 0.816·6-s − 1.13·7-s + 1/3·9-s + 0.632·10-s − 1.50·11-s + 0.577·12-s + 0.277·13-s − 1.60·14-s + 0.258·15-s − 16-s + 1.21·17-s + 0.471·18-s + 0.458·19-s + 0.447·20-s − 0.654·21-s − 2.13·22-s − 0.208·23-s + 1/5·25-s + 0.392·26-s + 0.192·27-s − 1.13·28-s + 1.85·29-s + 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.479355493\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.479355493\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 11 T + p T^{2} \) |
| 97 | \( 1 + 9 T + p T^{2} \) |
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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
−12.70705052788882338740212030436, −12.12132680691425538467753717871, −10.48774420896020938364797406259, −9.740914180850663168496545923199, −8.434475207546954053625988897044, −7.07755893352212952629549495812, −5.93954304028653148352357105381, −5.02803177361324098534837817934, −3.48200443418719418963780366073, −2.70048152954381141744823279421,
2.70048152954381141744823279421, 3.48200443418719418963780366073, 5.02803177361324098534837817934, 5.93954304028653148352357105381, 7.07755893352212952629549495812, 8.434475207546954053625988897044, 9.740914180850663168496545923199, 10.48774420896020938364797406259, 12.12132680691425538467753717871, 12.70705052788882338740212030436
