| L(s) = 1 | − 20·5-s + 29·9-s − 130·11-s − 32·19-s + 275·25-s + 114·29-s − 516·31-s − 828·41-s − 580·45-s − 49·49-s + 2.60e3·55-s − 668·59-s − 112·61-s + 1.90e3·71-s + 742·79-s + 112·81-s − 1.76e3·89-s + 640·95-s − 3.77e3·99-s − 296·101-s − 882·109-s + 1.00e4·121-s − 3.00e3·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 1.07·9-s − 3.56·11-s − 0.386·19-s + 11/5·25-s + 0.729·29-s − 2.98·31-s − 3.15·41-s − 1.92·45-s − 1/7·49-s + 6.37·55-s − 1.47·59-s − 0.235·61-s + 3.18·71-s + 1.05·79-s + 0.153·81-s − 2.09·89-s + 0.691·95-s − 3.82·99-s − 0.291·101-s − 0.775·109-s + 7.52·121-s − 2.14·125-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.2015800808\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2015800808\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + 4 p T + p^{3} T^{2} \) |
| 7 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 29 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 65 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 25 p^{2} T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2943 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 16 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 10262 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 57 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 258 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 83350 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 414 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 78358 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 32085 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 280854 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 334 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 56 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 316370 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 952 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 526030 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 371 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 344826 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 880 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1767265 T^{2} + p^{6} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.89928307476750469289325702076, −12.42892862544223968384309061576, −12.19011073858879881937051519408, −11.07617083125897536776018907365, −11.05414029722190155414619668993, −10.43861800004756773864356056019, −10.14146581520490382240196363025, −9.318968355577315730719135062149, −8.387966488792073252608804890729, −8.120901256497019864912488842837, −7.71426450478175156934879921426, −7.18868134761318974028682398455, −6.75480742025140333203253458557, −5.35868251277349773651686351688, −5.16628278057892746381922314071, −4.47777255431891693914593129708, −3.59923161898129051916478279125, −3.02351752308181255462777051669, −1.98003598545674203770433284795, −0.21378649255835336944031159276,
0.21378649255835336944031159276, 1.98003598545674203770433284795, 3.02351752308181255462777051669, 3.59923161898129051916478279125, 4.47777255431891693914593129708, 5.16628278057892746381922314071, 5.35868251277349773651686351688, 6.75480742025140333203253458557, 7.18868134761318974028682398455, 7.71426450478175156934879921426, 8.120901256497019864912488842837, 8.387966488792073252608804890729, 9.318968355577315730719135062149, 10.14146581520490382240196363025, 10.43861800004756773864356056019, 11.05414029722190155414619668993, 11.07617083125897536776018907365, 12.19011073858879881937051519408, 12.42892862544223968384309061576, 12.89928307476750469289325702076
