| L(s) = 1 | − 2·3-s + 2·5-s − 2·7-s + 3·9-s − 4·11-s + 2·13-s − 4·15-s + 6·17-s − 2·19-s + 4·21-s + 2·23-s + 3·25-s − 4·27-s + 6·31-s + 8·33-s − 4·35-s − 2·37-s − 4·39-s − 4·41-s − 4·43-s + 6·45-s + 18·47-s + 6·49-s − 12·51-s + 20·53-s − 8·55-s + 4·57-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.894·5-s − 0.755·7-s + 9-s − 1.20·11-s + 0.554·13-s − 1.03·15-s + 1.45·17-s − 0.458·19-s + 0.872·21-s + 0.417·23-s + 3/5·25-s − 0.769·27-s + 1.07·31-s + 1.39·33-s − 0.676·35-s − 0.328·37-s − 0.640·39-s − 0.624·41-s − 0.609·43-s + 0.894·45-s + 2.62·47-s + 6/7·49-s − 1.68·51-s + 2.74·53-s − 1.07·55-s + 0.529·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3686400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3686400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.871566810\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.871566810\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 + 2 T - 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_4$ | \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 6 T + 26 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2 T + 30 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 6 T + 54 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 2 T + 58 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 18 T + 158 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 - 8 T + 94 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 18 T + 222 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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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
−9.356791594914383908210474172565, −9.122500641723497903450375439740, −8.748323782677769810330679160595, −8.143208932893733349944922815051, −7.78009946351911600510911295873, −7.39536996372988463250358628039, −6.86663253534215083547353363129, −6.60336096654987276107479730905, −6.10346534127001598548807155674, −5.78356093226106515295114625768, −5.45397693751066073370802890689, −5.19060275857337787418167720091, −4.61313099545740155348448603371, −4.15959188512280780902083877018, −3.35398486943687777956529763738, −3.25433193910272785317620247166, −2.31308599493634910666141141733, −2.09642406181533508831481710746, −0.979120241433238298053492313580, −0.68522397140095519720806648681,
0.68522397140095519720806648681, 0.979120241433238298053492313580, 2.09642406181533508831481710746, 2.31308599493634910666141141733, 3.25433193910272785317620247166, 3.35398486943687777956529763738, 4.15959188512280780902083877018, 4.61313099545740155348448603371, 5.19060275857337787418167720091, 5.45397693751066073370802890689, 5.78356093226106515295114625768, 6.10346534127001598548807155674, 6.60336096654987276107479730905, 6.86663253534215083547353363129, 7.39536996372988463250358628039, 7.78009946351911600510911295873, 8.143208932893733349944922815051, 8.748323782677769810330679160595, 9.122500641723497903450375439740, 9.356791594914383908210474172565
