| L(s) = 1 | − 7-s + 3·11-s − 13-s + 12·17-s − 8·19-s + 3·23-s + 3·29-s − 5·31-s − 4·37-s + 3·41-s − 43-s + 9·47-s + 7·49-s − 12·53-s − 3·59-s + 13·61-s − 7·67-s + 24·71-s + 20·73-s − 3·77-s − 11·79-s + 9·83-s − 12·89-s + 91-s + 11·97-s + 15·101-s − 7·103-s + ⋯ |
| L(s) = 1 | − 0.377·7-s + 0.904·11-s − 0.277·13-s + 2.91·17-s − 1.83·19-s + 0.625·23-s + 0.557·29-s − 0.898·31-s − 0.657·37-s + 0.468·41-s − 0.152·43-s + 1.31·47-s + 49-s − 1.64·53-s − 0.390·59-s + 1.66·61-s − 0.855·67-s + 2.84·71-s + 2.34·73-s − 0.341·77-s − 1.23·79-s + 0.987·83-s − 1.27·89-s + 0.104·91-s + 1.11·97-s + 1.49·101-s − 0.689·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7290000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7290000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.984741412\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.984741412\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 3 T - 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + T - 42 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 9 T + 34 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 7 T - 18 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 11 T + 42 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 9 T - 2 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 11 T + 24 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
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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
−8.955093213528559743554773459310, −8.700059130137567478704344729243, −8.177541086239667217473287099925, −8.059132000704741858442902833906, −7.30703586745034570431967184093, −7.26759846298857389407542883761, −6.79755045385266710969987275832, −6.29883229225748573767382195166, −5.86223857572940461729381863652, −5.77748498234514986135667973289, −4.96931322002226625047524600859, −4.92450715109776756216432577419, −4.15319126485916762715176826787, −3.79850725971616067001394431616, −3.32143267160992003618237791114, −3.14594194224366965848176516650, −2.19135555622565995720201579620, −1.99513772462782543979242594603, −1.05141052407344781931165861592, −0.67656355528586409631354146224,
0.67656355528586409631354146224, 1.05141052407344781931165861592, 1.99513772462782543979242594603, 2.19135555622565995720201579620, 3.14594194224366965848176516650, 3.32143267160992003618237791114, 3.79850725971616067001394431616, 4.15319126485916762715176826787, 4.92450715109776756216432577419, 4.96931322002226625047524600859, 5.77748498234514986135667973289, 5.86223857572940461729381863652, 6.29883229225748573767382195166, 6.79755045385266710969987275832, 7.26759846298857389407542883761, 7.30703586745034570431967184093, 8.059132000704741858442902833906, 8.177541086239667217473287099925, 8.700059130137567478704344729243, 8.955093213528559743554773459310
