| L(s) = 1 | + 2·5-s + 2·7-s + 8·11-s + 4·13-s + 10·17-s + 2·19-s − 4·25-s + 10·29-s − 4·31-s + 4·35-s + 4·37-s + 4·41-s − 4·43-s + 10·47-s + 3·49-s + 2·53-s + 16·55-s + 8·59-s − 12·61-s + 8·65-s + 4·67-s + 2·71-s + 16·77-s − 8·79-s − 6·83-s + 20·85-s + 12·89-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.755·7-s + 2.41·11-s + 1.10·13-s + 2.42·17-s + 0.458·19-s − 4/5·25-s + 1.85·29-s − 0.718·31-s + 0.676·35-s + 0.657·37-s + 0.624·41-s − 0.609·43-s + 1.45·47-s + 3/7·49-s + 0.274·53-s + 2.15·55-s + 1.04·59-s − 1.53·61-s + 0.992·65-s + 0.488·67-s + 0.237·71-s + 1.82·77-s − 0.900·79-s − 0.658·83-s + 2.16·85-s + 1.27·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91699776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91699776 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.204620087\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.204620087\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - 2 T + 8 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 10 T + 56 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 10 T + 80 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 10 T + 92 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 2 T + 32 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 8 T + 86 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 12 T + 110 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4 T + 126 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 2 T + 116 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 8 T + 126 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 6 T + 148 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 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
−7.82101754553764157746297941289, −7.56567840244980236279477988784, −7.14757172655553317620861811931, −6.82766382421861412503423129086, −6.36147310357871424783045138282, −6.08999287243871066865960338651, −5.90501916147573538269381378539, −5.61475450422330316634752626865, −5.07992508981003439914161783340, −4.91240642106074471171370359062, −4.15824968975389268927721359387, −4.06271904862023123956389879638, −3.54088591409844500854624800673, −3.51292159455694548582057254415, −2.66624203473833713799894779997, −2.51367681212303007485765341692, −1.59512035975531476826909318526, −1.49063448981941371951529877006, −1.12594340283790499975146702965, −0.78338574392585962006356418634,
0.78338574392585962006356418634, 1.12594340283790499975146702965, 1.49063448981941371951529877006, 1.59512035975531476826909318526, 2.51367681212303007485765341692, 2.66624203473833713799894779997, 3.51292159455694548582057254415, 3.54088591409844500854624800673, 4.06271904862023123956389879638, 4.15824968975389268927721359387, 4.91240642106074471171370359062, 5.07992508981003439914161783340, 5.61475450422330316634752626865, 5.90501916147573538269381378539, 6.08999287243871066865960338651, 6.36147310357871424783045138282, 6.82766382421861412503423129086, 7.14757172655553317620861811931, 7.56567840244980236279477988784, 7.82101754553764157746297941289
