| L(s) = 1 | + 5·2-s − 4-s − 58·5-s − 286·7-s + 25·8-s − 290·10-s + 242·11-s − 166·13-s − 1.43e3·14-s − 73·16-s + 800·17-s − 1.47e3·19-s + 58·20-s + 1.21e3·22-s + 3.37e3·23-s + 698·25-s − 830·26-s + 286·28-s − 6.60e3·29-s − 7.52e3·31-s − 6.29e3·32-s + 4.00e3·34-s + 1.65e4·35-s − 2.99e4·37-s − 7.38e3·38-s − 1.45e3·40-s + 5.78e3·41-s + ⋯ |
| L(s) = 1 | + 0.883·2-s − 0.0312·4-s − 1.03·5-s − 2.20·7-s + 0.138·8-s − 0.917·10-s + 0.603·11-s − 0.272·13-s − 1.94·14-s − 0.0712·16-s + 0.671·17-s − 0.937·19-s + 0.0324·20-s + 0.533·22-s + 1.32·23-s + 0.223·25-s − 0.240·26-s + 0.0689·28-s − 1.45·29-s − 1.40·31-s − 1.08·32-s + 0.593·34-s + 2.28·35-s − 3.59·37-s − 0.829·38-s − 0.143·40-s + 0.536·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9801 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9801 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 5 T + 13 p T^{2} - 5 p^{5} T^{3} + p^{10} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 58 T + 2666 T^{2} + 58 p^{5} T^{3} + p^{10} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 286 T + 49638 T^{2} + 286 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 166 T + 685578 T^{2} + 166 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 800 T + 2840414 T^{2} - 800 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 1476 T + 5490470 T^{2} + 1476 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 3370 T + 9784358 T^{2} - 3370 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 6600 T + 44401126 T^{2} + 6600 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 7528 T + 66189630 T^{2} + 7528 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 29916 T + 361611230 T^{2} + 29916 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 5780 T + 217632230 T^{2} - 5780 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 16656 T + 264340262 T^{2} + 16656 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 7850 T + 191750726 T^{2} + 7850 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 14178 T + 671305114 T^{2} + 14178 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 17300 T + 1408269110 T^{2} + 17300 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 2946 T + 1451133506 T^{2} + 2946 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 31336 T + 2492616438 T^{2} - 31336 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 33810 T + 3469543750 T^{2} - 33810 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 60644 T + 2552552022 T^{2} - 60644 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 1870 T + 2176543686 T^{2} - 1870 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 58296 T + 101571026 p T^{2} - 58296 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 92388 T + 3271275766 T^{2} + 92388 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 7120 T - 2888428386 T^{2} - 7120 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−12.58651233270413648155009196573, −12.49490107480554404470257820953, −11.88659848996773380570557615830, −11.03296711132702169116769874310, −10.73770716998413694745198259639, −9.945359298232763660747264942821, −9.272018085041959485870937312845, −9.083440140782951996034391299719, −8.195603649430960062568887109846, −7.36483369098179993731665941153, −6.78843185294373572905191810792, −6.57165746615799686687305656659, −5.43579015165024143836678590656, −5.03215328705966949122342433307, −3.81065455869255812422439185330, −3.71772727005613806213571283241, −3.13606339379531226478307639672, −1.74632912364525360646673306111, 0, 0,
1.74632912364525360646673306111, 3.13606339379531226478307639672, 3.71772727005613806213571283241, 3.81065455869255812422439185330, 5.03215328705966949122342433307, 5.43579015165024143836678590656, 6.57165746615799686687305656659, 6.78843185294373572905191810792, 7.36483369098179993731665941153, 8.195603649430960062568887109846, 9.083440140782951996034391299719, 9.272018085041959485870937312845, 9.945359298232763660747264942821, 10.73770716998413694745198259639, 11.03296711132702169116769874310, 11.88659848996773380570557615830, 12.49490107480554404470257820953, 12.58651233270413648155009196573
