| L(s) = 1 | − 18·9-s − 136·17-s + 484·25-s + 104·41-s − 972·49-s − 1.35e3·73-s + 243·81-s − 936·89-s − 712·97-s + 5.51e3·113-s + 4.52e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 2.44e3·153-s + 157-s + 163-s + 167-s + 5.65e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | − 2/3·9-s − 1.94·17-s + 3.87·25-s + 0.396·41-s − 2.83·49-s − 2.16·73-s + 1/3·81-s − 1.11·89-s − 0.745·97-s + 4.58·113-s + 3.39·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 1.29·153-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 2.57·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.732480290\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.732480290\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| good | 5 | $C_2^2$ | \( ( 1 - 242 T^{2} + p^{6} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 486 T^{2} + p^{6} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 2262 T^{2} + p^{6} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 2826 T^{2} + p^{6} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 2 p T + p^{3} T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - 11014 T^{2} + p^{6} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 20462 T^{2} + p^{6} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 8450 T^{2} + p^{6} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 47414 T^{2} + p^{6} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 27578 T^{2} + p^{6} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 26 T + p^{3} T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 95510 T^{2} + p^{6} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 88574 T^{2} + p^{6} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 166894 T^{2} + p^{6} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 278262 T^{2} + p^{6} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 86838 T^{2} + p^{6} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 207142 T^{2} + p^{6} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 604430 T^{2} + p^{6} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 338 T + p^{3} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 363350 T^{2} + p^{6} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 70278 T^{2} + p^{6} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 234 T + p^{3} T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 + 178 T + p^{3} T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.899503691105174176646368595144, −7.40226151026692747736564609205, −7.23753796132891260030602296272, −6.92835586582445081438932703400, −6.83248560405468646779664952422, −6.48318552662670867163721831558, −6.40113102970951909394524857088, −5.91478345948786207914828955378, −5.85020192010675186257053136414, −5.49283355215879112799614352490, −4.88424119895091956135116504667, −4.87793183203197561233041376106, −4.69010627665298541652862564958, −4.54507972618124683788052805180, −4.08496260060414511643724764386, −3.55137003746182330282803491242, −3.40875374196671540546513024122, −2.93015530507394179874910201207, −2.83489369265530440013514325255, −2.42598263830314613612738146260, −2.11915038568140354187990086097, −1.41718853309754787593893911810, −1.35427127267376563876911138928, −0.64700293773732759806390948661, −0.25486677170884438806167587978,
0.25486677170884438806167587978, 0.64700293773732759806390948661, 1.35427127267376563876911138928, 1.41718853309754787593893911810, 2.11915038568140354187990086097, 2.42598263830314613612738146260, 2.83489369265530440013514325255, 2.93015530507394179874910201207, 3.40875374196671540546513024122, 3.55137003746182330282803491242, 4.08496260060414511643724764386, 4.54507972618124683788052805180, 4.69010627665298541652862564958, 4.87793183203197561233041376106, 4.88424119895091956135116504667, 5.49283355215879112799614352490, 5.85020192010675186257053136414, 5.91478345948786207914828955378, 6.40113102970951909394524857088, 6.48318552662670867163721831558, 6.83248560405468646779664952422, 6.92835586582445081438932703400, 7.23753796132891260030602296272, 7.40226151026692747736564609205, 7.899503691105174176646368595144
