| L(s) = 1 | − 10·5-s + 14·7-s − 22·11-s + 24·13-s − 30·17-s + 32·19-s − 82·23-s − 38·25-s − 36·29-s + 112·31-s − 140·35-s + 48·37-s + 70·41-s + 40·43-s − 420·47-s + 147·49-s − 176·53-s + 220·55-s − 404·59-s − 156·61-s − 240·65-s − 4·67-s − 814·71-s − 216·73-s − 308·77-s + 1.58e3·79-s − 464·83-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.755·7-s − 0.603·11-s + 0.512·13-s − 0.428·17-s + 0.386·19-s − 0.743·23-s − 0.303·25-s − 0.230·29-s + 0.648·31-s − 0.676·35-s + 0.213·37-s + 0.266·41-s + 0.141·43-s − 1.30·47-s + 3/7·49-s − 0.456·53-s + 0.539·55-s − 0.891·59-s − 0.327·61-s − 0.457·65-s − 0.00729·67-s − 1.36·71-s − 0.346·73-s − 0.455·77-s + 2.25·79-s − 0.613·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4064256 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4064256 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 2 p T + 138 T^{2} + 2 p^{4} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 2 p T + 2646 T^{2} + 2 p^{4} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 24 T + 3990 T^{2} - 24 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 30 T + 9914 T^{2} + 30 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 32 T + 11782 T^{2} - 32 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 82 T + 24782 T^{2} + 82 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 36 T - 5698 T^{2} + 36 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 112 T + 53950 T^{2} - 112 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 48 T + 75030 T^{2} - 48 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 70 T + 138930 T^{2} - 70 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 40 T + 104614 T^{2} - 40 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 420 T + 93374 T^{2} + 420 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 176 T + 261110 T^{2} + 176 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 404 T + 451014 T^{2} + 404 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 156 T + 240846 T^{2} + 156 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 4 T + 478230 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 814 T + 878046 T^{2} + 814 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 216 T + 499806 T^{2} + 216 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 20 p T + 1543870 T^{2} - 20 p^{4} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 464 T + 1195206 T^{2} + 464 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 1158 T + 708226 T^{2} - 1158 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 1760 T + 2073118 T^{2} + 1760 p^{3} T^{3} + p^{6} T^{4} \) |
| 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
−8.425077745435722842926824862329, −8.099918591650315748817648159301, −7.84718503682267978036546827105, −7.69521654852664879023979262997, −7.03914157714482146379876299158, −6.75374087226237943604039092968, −6.04959764971623916316765115853, −5.97799978387022789884499266839, −5.30616845138106503132243771924, −4.92368345712595939426706844821, −4.38590236322994239344849232568, −4.28188935480198014207640242787, −3.42976660610154066405878671198, −3.40940626901013382646746274510, −2.54309258226003951480573266716, −2.18177039416412037657844654749, −1.45055853499535410592568307214, −1.04666453887660452133669179193, 0, 0,
1.04666453887660452133669179193, 1.45055853499535410592568307214, 2.18177039416412037657844654749, 2.54309258226003951480573266716, 3.40940626901013382646746274510, 3.42976660610154066405878671198, 4.28188935480198014207640242787, 4.38590236322994239344849232568, 4.92368345712595939426706844821, 5.30616845138106503132243771924, 5.97799978387022789884499266839, 6.04959764971623916316765115853, 6.75374087226237943604039092968, 7.03914157714482146379876299158, 7.69521654852664879023979262997, 7.84718503682267978036546827105, 8.099918591650315748817648159301, 8.425077745435722842926824862329
