| L(s) = 1 | + 6·5-s − 18·9-s + 84·11-s + 112·19-s + 146·25-s + 636·29-s − 104·31-s − 816·41-s − 108·45-s + 616·49-s + 504·55-s − 372·59-s + 680·61-s + 72·71-s + 760·79-s + 243·81-s − 2.23e3·89-s + 672·95-s − 1.51e3·99-s − 2.24e3·101-s + 1.32e3·109-s − 176·121-s + 2.28e3·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 0.536·5-s − 2/3·9-s + 2.30·11-s + 1.35·19-s + 1.16·25-s + 4.07·29-s − 0.602·31-s − 3.10·41-s − 0.357·45-s + 1.79·49-s + 1.23·55-s − 0.820·59-s + 1.42·61-s + 0.120·71-s + 1.08·79-s + 1/3·81-s − 2.65·89-s + 0.725·95-s − 1.53·99-s − 2.21·101-s + 1.16·109-s − 0.132·121-s + 1.63·125-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(7.195884273\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.195884273\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 - 6 T - 22 p T^{2} - 6 p^{3} T^{3} + p^{6} T^{4} \) |
| good | 7 | $D_4\times C_2$ | \( 1 - 88 p T^{2} + 316878 T^{4} - 88 p^{7} T^{6} + p^{12} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 42 T + 2734 T^{2} - 42 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 5008 T^{2} + 13678638 T^{4} - 5008 p^{6} T^{6} + p^{12} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 12400 T^{2} + 76051038 T^{4} - 12400 p^{6} T^{6} + p^{12} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 56 T + 8598 T^{2} - 56 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 46204 T^{2} + 828262758 T^{4} - 46204 p^{6} T^{6} + p^{12} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 318 T + 55978 T^{2} - 318 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 52 T + 58782 T^{2} + 52 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 96016 T^{2} + 4637534478 T^{4} - 96016 p^{6} T^{6} + p^{12} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 408 T + 177982 T^{2} + 408 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 121900 T^{2} + 14997346998 T^{4} - 121900 p^{6} T^{6} + p^{12} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 225580 T^{2} + 31469120358 T^{4} - 225580 p^{6} T^{6} + p^{12} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 376864 T^{2} + 68697431598 T^{4} - 376864 p^{6} T^{6} + p^{12} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 186 T + 419038 T^{2} + 186 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 340 T + 388398 T^{2} - 340 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 861340 T^{2} + 346621507638 T^{4} - 861340 p^{6} T^{6} + p^{12} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 36 T + 384046 T^{2} - 36 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 429844 T^{2} + 136740794118 T^{4} - 429844 p^{6} T^{6} + p^{12} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 380 T + 99678 T^{2} - 380 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 916108 T^{2} + 434315280918 T^{4} - 916108 p^{6} T^{6} + p^{12} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 1116 T + 1508758 T^{2} + 1116 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 2174980 T^{2} + 2500346420358 T^{4} - 2174980 p^{6} T^{6} + p^{12} T^{8} \) |
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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
−8.468806161975016542334420598607, −8.297004309622728362340769523241, −7.935695218876817928383217947314, −7.59665578715500085231648764630, −7.15884416398857355615678534789, −6.85021968071175186262019838903, −6.65974904373807112935210047952, −6.60246198968302584698964331251, −6.49657354461984138519113280230, −5.86767483691853404448270643801, −5.62369096353377263030426136308, −5.33714319863991330525691852953, −5.15600016401687371593053615653, −4.62509143407575368562767966912, −4.53301367468974616662527267733, −3.94421473548856485499658394446, −3.92118624279717952950188701975, −3.14984399947062200599464808055, −3.05741709596481929214501156422, −2.91314539523736732073359071230, −2.21989783275900112060770487565, −1.70917454443674480679747490419, −1.32985066520681568846770767027, −0.900694592795974047348870824294, −0.58162469147347738379805104202,
0.58162469147347738379805104202, 0.900694592795974047348870824294, 1.32985066520681568846770767027, 1.70917454443674480679747490419, 2.21989783275900112060770487565, 2.91314539523736732073359071230, 3.05741709596481929214501156422, 3.14984399947062200599464808055, 3.92118624279717952950188701975, 3.94421473548856485499658394446, 4.53301367468974616662527267733, 4.62509143407575368562767966912, 5.15600016401687371593053615653, 5.33714319863991330525691852953, 5.62369096353377263030426136308, 5.86767483691853404448270643801, 6.49657354461984138519113280230, 6.60246198968302584698964331251, 6.65974904373807112935210047952, 6.85021968071175186262019838903, 7.15884416398857355615678534789, 7.59665578715500085231648764630, 7.935695218876817928383217947314, 8.297004309622728362340769523241, 8.468806161975016542334420598607
