| L(s) = 1 | − 4·2-s + 12·4-s − 2·5-s + 12·7-s − 32·8-s + 8·10-s + 12·11-s − 2·13-s − 48·14-s + 80·16-s − 20·17-s − 24·20-s − 48·22-s + 48·23-s + 25·25-s + 8·26-s + 144·28-s − 26·29-s + 12·31-s − 192·32-s + 80·34-s − 24·35-s + 52·37-s + 64·40-s + 58·41-s + 84·43-s + 144·44-s + ⋯ |
| L(s) = 1 | − 2·2-s + 3·4-s − 2/5·5-s + 12/7·7-s − 4·8-s + 4/5·10-s + 1.09·11-s − 0.153·13-s − 3.42·14-s + 5·16-s − 1.17·17-s − 6/5·20-s − 2.18·22-s + 2.08·23-s + 25-s + 4/13·26-s + 36/7·28-s − 0.896·29-s + 0.387·31-s − 6·32-s + 2.35·34-s − 0.685·35-s + 1.40·37-s + 8/5·40-s + 1.41·41-s + 1.95·43-s + 3.27·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.292980712\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.292980712\) |
| \(L(2)\) |
|
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 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T - 21 T^{2} + 2 p^{2} T^{3} + p^{4} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 12 T + 97 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 12 T + 169 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 2 T - 165 T^{2} + 2 p^{2} T^{3} + p^{4} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 10 T + p^{2} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 290 T^{2} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 48 T + 1297 T^{2} - 48 p^{2} T^{3} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 26 T - 165 T^{2} + 26 p^{2} T^{3} + p^{4} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 12 T + 1009 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 26 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 58 T + 1683 T^{2} - 58 p^{2} T^{3} + p^{4} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 84 T + 4201 T^{2} - 84 p^{2} T^{3} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 120 T + 7009 T^{2} - 120 p^{2} T^{3} + p^{4} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 74 T + p^{2} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 156 T + 11593 T^{2} - 156 p^{2} T^{3} + p^{4} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 26 T - 3045 T^{2} + 26 p^{2} T^{3} + p^{4} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 12 T + 4537 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 46 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 204 T + 20113 T^{2} + 204 p^{2} T^{3} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 84 T + 9241 T^{2} - 84 p^{2} T^{3} + p^{4} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 82 T + p^{2} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 167 T + p^{2} T^{2} )( 1 + 169 T + p^{2} T^{2} ) \) |
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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
−11.32342697177035376329168849056, −11.18956949299456443333777152275, −10.57343023762388450605430204160, −10.52662376259500383810063910299, −9.391496220202918459623851581246, −9.281168964452570550294235284545, −8.679557167601075018197975064695, −8.642791004353672293634621154149, −7.892691991204794734106368884477, −7.42030350416900119641351903994, −6.97274054304662647831914561110, −6.82363986545806079866501190318, −5.67796776859651319830242165245, −5.56578637096804299264096363786, −4.34089289880176737539494452883, −4.03746810529989418617691615636, −2.55907290003164837548475178786, −2.50222174393103856774586475045, −1.18496665722771437086411830623, −0.933804477186919192932926758822,
0.933804477186919192932926758822, 1.18496665722771437086411830623, 2.50222174393103856774586475045, 2.55907290003164837548475178786, 4.03746810529989418617691615636, 4.34089289880176737539494452883, 5.56578637096804299264096363786, 5.67796776859651319830242165245, 6.82363986545806079866501190318, 6.97274054304662647831914561110, 7.42030350416900119641351903994, 7.892691991204794734106368884477, 8.642791004353672293634621154149, 8.679557167601075018197975064695, 9.281168964452570550294235284545, 9.391496220202918459623851581246, 10.52662376259500383810063910299, 10.57343023762388450605430204160, 11.18956949299456443333777152275, 11.32342697177035376329168849056
