| L(s) = 1 | + 2-s + 3·3-s + 3·5-s + 3·6-s + 7-s − 8-s + 6·9-s + 3·10-s − 6·11-s + 2·13-s + 14-s + 9·15-s − 16-s + 6·17-s + 6·18-s − 14·19-s + 3·21-s − 6·22-s + 6·23-s − 3·24-s + 25-s + 2·26-s + 9·27-s + 15·29-s + 9·30-s − 31-s − 18·33-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.73·3-s + 1.34·5-s + 1.22·6-s + 0.377·7-s − 0.353·8-s + 2·9-s + 0.948·10-s − 1.80·11-s + 0.554·13-s + 0.267·14-s + 2.32·15-s − 1/4·16-s + 1.45·17-s + 1.41·18-s − 3.21·19-s + 0.654·21-s − 1.27·22-s + 1.25·23-s − 0.612·24-s + 1/5·25-s + 0.392·26-s + 1.73·27-s + 2.78·29-s + 1.64·30-s − 0.179·31-s − 3.13·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.825450640\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.825450640\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 7 | $C_2$ | \( 1 - T + p T^{2} \) |
| 13 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 3 T + 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 15 T + 104 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + T - 30 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 9 T + 68 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + T - 42 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 3 T + 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 3 T + 56 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 18 T + 167 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 119 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 131 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 9 T + 10 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 13 T + 96 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + T - 78 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 154 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 12 T + 137 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.74372793858165603646458344877, −10.40599021814903233494159102102, −10.15586007818550702984468876163, −9.834938720355728077435155031090, −9.107092658571678320926294911471, −8.739548404838418587888437191782, −8.398047365693594049626957191404, −8.058060164040389894862853771617, −7.64584527518453847966192925392, −6.88711993768394632597413624108, −6.41281119721900638368003076189, −5.94003893984480475376863647178, −5.41539999020835388894773984016, −4.67657396129137277749226430290, −4.54881747004246941964133027837, −3.69958905311988802521932387899, −3.05817281814675295034763128246, −2.45498288846257318446906397101, −2.30842711932641985125490006132, −1.31280746891108112408432581906,
1.31280746891108112408432581906, 2.30842711932641985125490006132, 2.45498288846257318446906397101, 3.05817281814675295034763128246, 3.69958905311988802521932387899, 4.54881747004246941964133027837, 4.67657396129137277749226430290, 5.41539999020835388894773984016, 5.94003893984480475376863647178, 6.41281119721900638368003076189, 6.88711993768394632597413624108, 7.64584527518453847966192925392, 8.058060164040389894862853771617, 8.398047365693594049626957191404, 8.739548404838418587888437191782, 9.107092658571678320926294911471, 9.834938720355728077435155031090, 10.15586007818550702984468876163, 10.40599021814903233494159102102, 10.74372793858165603646458344877
