L(s) = 1 | − 2·3-s − 2·5-s + 2·7-s + 3·9-s − 4·13-s + 4·15-s + 4·17-s − 4·21-s + 3·25-s − 4·27-s + 4·29-s + 8·31-s − 4·35-s − 12·37-s + 8·39-s + 4·41-s + 8·43-s − 6·45-s − 16·47-s + 3·49-s − 8·51-s − 4·53-s + 8·59-s − 12·61-s + 6·63-s + 8·65-s + 8·67-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 0.894·5-s + 0.755·7-s + 9-s − 1.10·13-s + 1.03·15-s + 0.970·17-s − 0.872·21-s + 3/5·25-s − 0.769·27-s + 0.742·29-s + 1.43·31-s − 0.676·35-s − 1.97·37-s + 1.28·39-s + 0.624·41-s + 1.21·43-s − 0.894·45-s − 2.33·47-s + 3/7·49-s − 1.12·51-s − 0.549·53-s + 1.04·59-s − 1.53·61-s + 0.755·63-s + 0.992·65-s + 0.977·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45158400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45158400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.405384423\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.405384423\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
good | 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_4$ | \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 12 T + 78 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 16 T + 126 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 + 12 T + 126 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 8 T + 118 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
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
−7.982369407177063122723078947725, −7.953056124101357801219717123241, −7.30503799426494971263094718696, −7.19558709263979623534900079399, −6.70987150077684420608156129207, −6.53871101832283354297209600045, −6.01919851829340760619482452775, −5.60159504054638159408297301681, −5.23370760531694699447169231722, −4.97553101726940702119317904765, −4.55284017186599483225658868334, −4.50714788696721364699416077211, −3.69372139660429740121025821035, −3.65360593517275508357171536654, −2.84602279402718611290627211688, −2.68257461223782511830123581662, −1.84998376772649747350168218787, −1.52147072289896625681434426791, −0.811056742019968578546735780819, −0.43415504156937784028334735364,
0.43415504156937784028334735364, 0.811056742019968578546735780819, 1.52147072289896625681434426791, 1.84998376772649747350168218787, 2.68257461223782511830123581662, 2.84602279402718611290627211688, 3.65360593517275508357171536654, 3.69372139660429740121025821035, 4.50714788696721364699416077211, 4.55284017186599483225658868334, 4.97553101726940702119317904765, 5.23370760531694699447169231722, 5.60159504054638159408297301681, 6.01919851829340760619482452775, 6.53871101832283354297209600045, 6.70987150077684420608156129207, 7.19558709263979623534900079399, 7.30503799426494971263094718696, 7.953056124101357801219717123241, 7.982369407177063122723078947725