| L(s) = 1 | + 20·25-s + 26·49-s − 28·73-s + 76·97-s + 44·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯ |
| L(s) = 1 | + 4·25-s + 26/7·49-s − 3.27·73-s + 7.71·97-s + 4·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.153·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.479154360\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.479154360\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 7 | $C_2^2$ | \( ( 1 - 13 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2}( 1 + 5 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 + 11 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2}( 1 + 11 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )^{2}( 1 + T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 109 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 131 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - 19 T + p T^{2} )^{4} \) |
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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
−6.54360432878599849149266302293, −6.46447838581294152017691234573, −6.17466505451193222414039162449, −6.14055620025216768310432286080, −5.72643963790198297914648659281, −5.61565957265397899486010558271, −5.22193300531168785282336725978, −5.09988964480557560625905683404, −5.07887794436527299591140438246, −4.51152190074165218393384697165, −4.46014542533290480136668642150, −4.33302742211308525684304569217, −4.14657992536624399655077743511, −3.69616270441936405010368922630, −3.23215322524659702930540244803, −3.18352645628061160430463937916, −3.15801042324966766812408215396, −2.83010831875980841502397348953, −2.35307483015941522859370350550, −2.07808415771153746447899347587, −2.06808796982476330780568652731, −1.46644145677579914029027627327, −0.935656432774119223869585826208, −0.878657737742519940725745708193, −0.53684032415349759769797296609,
0.53684032415349759769797296609, 0.878657737742519940725745708193, 0.935656432774119223869585826208, 1.46644145677579914029027627327, 2.06808796982476330780568652731, 2.07808415771153746447899347587, 2.35307483015941522859370350550, 2.83010831875980841502397348953, 3.15801042324966766812408215396, 3.18352645628061160430463937916, 3.23215322524659702930540244803, 3.69616270441936405010368922630, 4.14657992536624399655077743511, 4.33302742211308525684304569217, 4.46014542533290480136668642150, 4.51152190074165218393384697165, 5.07887794436527299591140438246, 5.09988964480557560625905683404, 5.22193300531168785282336725978, 5.61565957265397899486010558271, 5.72643963790198297914648659281, 6.14055620025216768310432286080, 6.17466505451193222414039162449, 6.46447838581294152017691234573, 6.54360432878599849149266302293
