| L(s) = 1 | − 2·3-s + 2·5-s + 2·9-s + 6·11-s + 6·13-s − 4·15-s + 2·19-s + 16·23-s − 25-s − 6·27-s + 6·29-s − 12·33-s + 6·37-s − 12·39-s − 6·43-s + 4·45-s − 14·49-s − 18·53-s + 12·55-s − 4·57-s − 18·59-s − 10·61-s + 12·65-s + 6·67-s − 32·69-s + 12·73-s + 2·75-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.894·5-s + 2/3·9-s + 1.80·11-s + 1.66·13-s − 1.03·15-s + 0.458·19-s + 3.33·23-s − 1/5·25-s − 1.15·27-s + 1.11·29-s − 2.08·33-s + 0.986·37-s − 1.92·39-s − 0.914·43-s + 0.596·45-s − 2·49-s − 2.47·53-s + 1.61·55-s − 0.529·57-s − 2.34·59-s − 1.28·61-s + 1.48·65-s + 0.733·67-s − 3.85·69-s + 1.40·73-s + 0.230·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.581314962\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.581314962\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 90 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
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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
−11.50215152050005401709128680070, −11.40132360649148055994037119652, −11.05951390877709589172759414747, −10.78648730050423430781798730642, −9.835349881938172990073592918218, −9.643367671978985906779767123385, −9.127905672266235030647007906370, −8.789685722396959186889625466541, −8.133244739273812371385877596602, −7.39783820172731734990049752671, −6.72399724381487767526207173912, −6.41768309490026811865075907519, −6.10328301280886839159462132034, −5.60110123390204556726549497183, −4.71413021304624055773597026749, −4.61808802857359198520093291147, −3.42622832915344619222425503842, −3.12641082032681766964758731637, −1.48006086817518078236915258391, −1.25589686520630742536294150701,
1.25589686520630742536294150701, 1.48006086817518078236915258391, 3.12641082032681766964758731637, 3.42622832915344619222425503842, 4.61808802857359198520093291147, 4.71413021304624055773597026749, 5.60110123390204556726549497183, 6.10328301280886839159462132034, 6.41768309490026811865075907519, 6.72399724381487767526207173912, 7.39783820172731734990049752671, 8.133244739273812371385877596602, 8.789685722396959186889625466541, 9.127905672266235030647007906370, 9.643367671978985906779767123385, 9.835349881938172990073592918218, 10.78648730050423430781798730642, 11.05951390877709589172759414747, 11.40132360649148055994037119652, 11.50215152050005401709128680070
