| L(s) = 1 | + 2·3-s − 7-s + 3·9-s − 4·11-s + 4·13-s − 3·17-s − 2·21-s + 3·23-s + 10·27-s − 12·29-s − 9·31-s − 8·33-s + 8·39-s + 10·41-s + 12·43-s + 9·47-s − 6·49-s − 6·51-s − 6·53-s − 8·59-s − 8·61-s − 3·63-s + 14·67-s + 6·69-s + 22·71-s − 2·73-s + 4·77-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.377·7-s + 9-s − 1.20·11-s + 1.10·13-s − 0.727·17-s − 0.436·21-s + 0.625·23-s + 1.92·27-s − 2.22·29-s − 1.61·31-s − 1.39·33-s + 1.28·39-s + 1.56·41-s + 1.82·43-s + 1.31·47-s − 6/7·49-s − 0.840·51-s − 0.824·53-s − 1.04·59-s − 1.02·61-s − 0.377·63-s + 1.71·67-s + 0.722·69-s + 2.61·71-s − 0.234·73-s + 0.455·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.907529322\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.907529322\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T + T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 4 T + 5 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 9 T + 50 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 9 T + 34 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 8 T + 5 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 14 T + 129 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 9 T + 2 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 11 T + 32 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
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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
−9.442230302341272876940673159998, −9.316402721314430828805157942795, −8.911438316034635997633731928727, −8.828492515546851920530589745748, −7.963149347942184417149369613609, −7.88581045624189820291944312690, −7.42881538663985532489250437751, −7.13921398510323406508806801667, −6.48802365056284018082708748710, −6.10610431854181666728485279323, −5.65265750840310458565433198325, −5.13849713972030019075298364589, −4.65881410784362455438298351545, −4.06024491269712619656586931581, −3.59879437782135210841045002144, −3.33129490705527477394992038943, −2.53047616064698233314009098492, −2.31856820978026890140390153273, −1.58717608591610793170014415323, −0.64256761870295660791180051862,
0.64256761870295660791180051862, 1.58717608591610793170014415323, 2.31856820978026890140390153273, 2.53047616064698233314009098492, 3.33129490705527477394992038943, 3.59879437782135210841045002144, 4.06024491269712619656586931581, 4.65881410784362455438298351545, 5.13849713972030019075298364589, 5.65265750840310458565433198325, 6.10610431854181666728485279323, 6.48802365056284018082708748710, 7.13921398510323406508806801667, 7.42881538663985532489250437751, 7.88581045624189820291944312690, 7.963149347942184417149369613609, 8.828492515546851920530589745748, 8.911438316034635997633731928727, 9.316402721314430828805157942795, 9.442230302341272876940673159998
