| L(s) = 1 | + 2·2-s + 2·3-s + 3·4-s + 4·6-s + 4·8-s + 3·9-s − 2·11-s + 6·12-s + 2·13-s + 5·16-s + 2·17-s + 6·18-s + 6·19-s − 4·22-s − 4·23-s + 8·24-s − 8·25-s + 4·26-s + 4·27-s + 10·29-s + 8·31-s + 6·32-s − 4·33-s + 4·34-s + 9·36-s + 4·37-s + 12·38-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s + 3/2·4-s + 1.63·6-s + 1.41·8-s + 9-s − 0.603·11-s + 1.73·12-s + 0.554·13-s + 5/4·16-s + 0.485·17-s + 1.41·18-s + 1.37·19-s − 0.852·22-s − 0.834·23-s + 1.63·24-s − 8/5·25-s + 0.784·26-s + 0.769·27-s + 1.85·29-s + 1.43·31-s + 1.06·32-s − 0.696·33-s + 0.685·34-s + 3/2·36-s + 0.657·37-s + 1.94·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14607684 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14607684 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(15.77541511\) |
| \(L(\frac12)\) |
\(\approx\) |
\(15.77541511\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 2 T + 21 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 2 T + p T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 + 4 T + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 28 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 12 T + 116 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 82 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 - 2 T - 21 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 18 T + 197 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 2 T + 105 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 18 T + 183 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 - 12 T + 132 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 4 T + 34 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 4 T + 138 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 4 T + 164 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 4 T + 100 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.400251344616668488071078817234, −8.255778077666497004749933639107, −7.75434901552065513055240273509, −7.72689131307410011798144234982, −7.15945700935716349767501988009, −6.95637270481319983302452012980, −6.19415660202671647117484980075, −6.10927765434231720498422430719, −5.56691224608062999877867274729, −5.46150629397672533724213061011, −4.69006356954258470328763996880, −4.43417419444974324714972749693, −3.89842587071899771378089064077, −3.89130447155569416007337386990, −3.15891926873810763346200946546, −2.81721772140154688142432070458, −2.37753522456644293595311079088, −2.24084956120762708570795613312, −1.12365453925280019383108899458, −1.00598125479172727285268592316,
1.00598125479172727285268592316, 1.12365453925280019383108899458, 2.24084956120762708570795613312, 2.37753522456644293595311079088, 2.81721772140154688142432070458, 3.15891926873810763346200946546, 3.89130447155569416007337386990, 3.89842587071899771378089064077, 4.43417419444974324714972749693, 4.69006356954258470328763996880, 5.46150629397672533724213061011, 5.56691224608062999877867274729, 6.10927765434231720498422430719, 6.19415660202671647117484980075, 6.95637270481319983302452012980, 7.15945700935716349767501988009, 7.72689131307410011798144234982, 7.75434901552065513055240273509, 8.255778077666497004749933639107, 8.400251344616668488071078817234
