| L(s) = 1 | + 2-s − 7-s − 8-s + 3·11-s − 7·13-s − 14-s − 16-s + 3·17-s − 5·19-s + 3·22-s + 6·23-s − 10·25-s − 7·26-s − 3·29-s − 8·31-s + 3·34-s + 4·37-s − 5·38-s − 3·41-s − 8·43-s + 6·46-s − 18·47-s − 10·50-s + 18·53-s + 56-s − 3·58-s − 6·59-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.377·7-s − 0.353·8-s + 0.904·11-s − 1.94·13-s − 0.267·14-s − 1/4·16-s + 0.727·17-s − 1.14·19-s + 0.639·22-s + 1.25·23-s − 2·25-s − 1.37·26-s − 0.557·29-s − 1.43·31-s + 0.514·34-s + 0.657·37-s − 0.811·38-s − 0.468·41-s − 1.21·43-s + 0.884·46-s − 2.62·47-s − 1.41·50-s + 2.47·53-s + 0.133·56-s − 0.393·58-s − 0.781·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2683044 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2683044 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.005314210\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.005314210\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T + T^{2} \) |
| 13 | $C_2$ | \( 1 + 7 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 5 T + 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 4 T - 21 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 6 T - 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 5 T - 36 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 14 T + 129 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 6 T - 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 9 T - 8 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 8 T - 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−9.653442464396652947724529187769, −9.071683161936398029196703161524, −9.051297610560610383708346398818, −8.390590308237013648968201641780, −7.88667520343074473433397400528, −7.45813232339023579332972048810, −7.25860721786543634265575165528, −6.55542243702444089986673865799, −6.52135518519726336874058057142, −5.69922858977973432158621707153, −5.60276589977837162017963605570, −4.95931159244578837584615899882, −4.67141318112189569278912530795, −4.03128775332095571184509956250, −3.84153033207409697659694930803, −3.03596539766522234745644613513, −2.91722754221176858081016022074, −1.81307851726224342351687968094, −1.79733926639099007997162596308, −0.32319354833279571260236291351,
0.32319354833279571260236291351, 1.79733926639099007997162596308, 1.81307851726224342351687968094, 2.91722754221176858081016022074, 3.03596539766522234745644613513, 3.84153033207409697659694930803, 4.03128775332095571184509956250, 4.67141318112189569278912530795, 4.95931159244578837584615899882, 5.60276589977837162017963605570, 5.69922858977973432158621707153, 6.52135518519726336874058057142, 6.55542243702444089986673865799, 7.25860721786543634265575165528, 7.45813232339023579332972048810, 7.88667520343074473433397400528, 8.390590308237013648968201641780, 9.051297610560610383708346398818, 9.071683161936398029196703161524, 9.653442464396652947724529187769
