| L(s) = 1 | − 2·3-s − 2·4-s + 3·9-s + 4·11-s + 4·12-s − 6·13-s − 4·17-s + 2·19-s + 4·23-s − 4·27-s + 8·29-s + 6·31-s − 8·33-s − 6·36-s + 14·37-s + 12·39-s + 4·41-s + 18·43-s − 8·44-s − 4·47-s + 8·51-s + 12·52-s − 8·53-s − 4·57-s − 8·61-s + 8·64-s + 2·67-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 4-s + 9-s + 1.20·11-s + 1.15·12-s − 1.66·13-s − 0.970·17-s + 0.458·19-s + 0.834·23-s − 0.769·27-s + 1.48·29-s + 1.07·31-s − 1.39·33-s − 36-s + 2.30·37-s + 1.92·39-s + 0.624·41-s + 2.74·43-s − 1.20·44-s − 0.583·47-s + 1.12·51-s + 1.66·52-s − 1.09·53-s − 0.529·57-s − 1.02·61-s + 64-s + 0.244·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13505625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13505625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.438105391\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.438105391\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 4 T + 24 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 6 T + 33 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 20 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2 T + 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 4 T + 32 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 8 T + 56 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 6 T + 63 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 14 T + 121 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 4 T + 68 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 18 T + 165 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 4 T - 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 8 T + 114 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 116 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 8 T + 66 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 2 T - 27 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 4 T + 144 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 10 T + 121 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 2 T + 127 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 8 T + 180 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 16 T + 224 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 16 T + 250 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 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
−8.863701418106296579101451158336, −8.400519321894345012296143071478, −7.76746313552372222434164172917, −7.69947008765459759289671968342, −7.22598224244007192603762208176, −6.64425396225236903391838791760, −6.52319003675254118251816681949, −6.24287521576465307735714915903, −5.57062530098527140489646703832, −5.41156057170442542516528843876, −4.66235134942954129317589692882, −4.59628163177004474227116475366, −4.29160748230650116852155823520, −4.13391911713437633549715854212, −3.10913486701807346138458309043, −2.81403547287180611764521706507, −2.29364946735546596834517002455, −1.54299918890633481153414445335, −0.67992057839945025960631793873, −0.67462337375577879961019379796,
0.67462337375577879961019379796, 0.67992057839945025960631793873, 1.54299918890633481153414445335, 2.29364946735546596834517002455, 2.81403547287180611764521706507, 3.10913486701807346138458309043, 4.13391911713437633549715854212, 4.29160748230650116852155823520, 4.59628163177004474227116475366, 4.66235134942954129317589692882, 5.41156057170442542516528843876, 5.57062530098527140489646703832, 6.24287521576465307735714915903, 6.52319003675254118251816681949, 6.64425396225236903391838791760, 7.22598224244007192603762208176, 7.69947008765459759289671968342, 7.76746313552372222434164172917, 8.400519321894345012296143071478, 8.863701418106296579101451158336
