| L(s) = 1 | + 2-s − 4-s + 2·7-s − 3·8-s + 11-s + 2·14-s − 16-s + 4·17-s + 22-s − 9·23-s − 2·28-s − 2·29-s + 8·31-s + 5·32-s + 4·34-s + 37-s + 6·41-s − 8·43-s − 44-s − 9·46-s + 8·47-s − 3·49-s − 2·53-s − 6·56-s − 2·58-s + 12·59-s − 7·61-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s + 0.755·7-s − 1.06·8-s + 0.301·11-s + 0.534·14-s − 1/4·16-s + 0.970·17-s + 0.213·22-s − 1.87·23-s − 0.377·28-s − 0.371·29-s + 1.43·31-s + 0.883·32-s + 0.685·34-s + 0.164·37-s + 0.937·41-s − 1.21·43-s − 0.150·44-s − 1.32·46-s + 1.16·47-s − 3/7·49-s − 0.274·53-s − 0.801·56-s − 0.262·58-s + 1.56·59-s − 0.896·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.742191106\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.742191106\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−14.57872542562440, −14.42179623483532, −13.70377207871102, −13.57220059209087, −12.70627570476114, −12.27219659831915, −11.78361260340669, −11.47915745352514, −10.61067643948030, −9.948156074089595, −9.725956952725656, −8.878294349053597, −8.403551909521396, −7.886380641934358, −7.401889547227997, −6.356311392535621, −6.071676046517291, −5.380659844197855, −4.872406474479290, −4.209496435209939, −3.818676450982252, −3.063312281519652, −2.299524015135654, −1.445883007588736, −0.5666306589136752,
0.5666306589136752, 1.445883007588736, 2.299524015135654, 3.063312281519652, 3.818676450982252, 4.209496435209939, 4.872406474479290, 5.380659844197855, 6.071676046517291, 6.356311392535621, 7.401889547227997, 7.886380641934358, 8.403551909521396, 8.878294349053597, 9.725956952725656, 9.948156074089595, 10.61067643948030, 11.47915745352514, 11.78361260340669, 12.27219659831915, 12.70627570476114, 13.57220059209087, 13.70377207871102, 14.42179623483532, 14.57872542562440
