| L(s) = 1 | − 8·11-s + 4·19-s − 8·29-s − 16·31-s − 8·41-s + 4·49-s − 32·59-s − 32·61-s + 32·71-s + 16·79-s + 14·81-s − 8·89-s + 16·101-s + 24·109-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | − 2.41·11-s + 0.917·19-s − 1.48·29-s − 2.87·31-s − 1.24·41-s + 4/7·49-s − 4.16·59-s − 4.09·61-s + 3.79·71-s + 1.80·79-s + 14/9·81-s − 0.847·89-s + 1.59·101-s + 2.29·109-s − 0.363·121-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 + 8/13·169-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.135861415\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.135861415\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 3 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} ) \) |
| 7 | $C_4\times C_2$ | \( 1 - 4 T^{2} - 26 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 13 | $D_4\times C_2$ | \( 1 - 8 T^{2} + 66 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 44 T^{2} + 934 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 + 4 T - 10 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 40 T^{2} + 546 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 148 T^{2} + 9046 T^{4} - 148 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 164 T^{2} + 11014 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 104 T^{2} + 7522 T^{4} - 104 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 16 T + 150 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 16 T + 154 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 256 T^{2} + 25330 T^{4} - 256 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 16 T + 198 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 1734 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 8 T + 142 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 68 T^{2} + 12886 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 4 T + 110 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 280 T^{2} + 35826 T^{4} - 280 p^{2} T^{6} + p^{4} T^{8} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.31122210658978051780581285669, −6.30208408258863143993010291043, −6.30083966969604342660653094607, −5.80011625532092285750856883137, −5.55695153622371043713285044332, −5.51093407788726407953362736484, −5.47138925203447059176193359816, −4.83156699223862316259037237170, −4.82235230169950396525764833131, −4.81851264190098231838596317870, −4.70788287172161161676475745778, −4.02068940598534697596547161893, −3.89838532985512108090909406237, −3.56433024233249576319646014572, −3.31064518014614791860158522328, −3.29583035949237515835783238268, −3.08389291244899957566431123968, −2.60593289889015598548523966471, −2.35724105000585277455318473274, −2.15920137626435536748547054814, −1.70306319486223363962787945059, −1.69876894568532541943955976164, −1.26901289972120952857082027657, −0.43965891675723827178753030623, −0.32077540743513859221672103138,
0.32077540743513859221672103138, 0.43965891675723827178753030623, 1.26901289972120952857082027657, 1.69876894568532541943955976164, 1.70306319486223363962787945059, 2.15920137626435536748547054814, 2.35724105000585277455318473274, 2.60593289889015598548523966471, 3.08389291244899957566431123968, 3.29583035949237515835783238268, 3.31064518014614791860158522328, 3.56433024233249576319646014572, 3.89838532985512108090909406237, 4.02068940598534697596547161893, 4.70788287172161161676475745778, 4.81851264190098231838596317870, 4.82235230169950396525764833131, 4.83156699223862316259037237170, 5.47138925203447059176193359816, 5.51093407788726407953362736484, 5.55695153622371043713285044332, 5.80011625532092285750856883137, 6.30083966969604342660653094607, 6.30208408258863143993010291043, 6.31122210658978051780581285669
