| L(s) = 1 | − 3-s + 5-s − 2·7-s − 2·9-s + 6·13-s − 15-s + 6·19-s + 2·21-s − 6·25-s + 2·27-s − 5·31-s − 2·35-s − 2·37-s − 6·39-s + 3·41-s + 11·43-s − 2·45-s + 2·47-s + 3·49-s − 11·53-s − 6·57-s + 4·59-s + 5·61-s + 4·63-s + 6·65-s − 7·67-s − 6·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.755·7-s − 2/3·9-s + 1.66·13-s − 0.258·15-s + 1.37·19-s + 0.436·21-s − 6/5·25-s + 0.384·27-s − 0.898·31-s − 0.338·35-s − 0.328·37-s − 0.960·39-s + 0.468·41-s + 1.67·43-s − 0.298·45-s + 0.291·47-s + 3/7·49-s − 1.51·53-s − 0.794·57-s + 0.520·59-s + 0.640·61-s + 0.503·63-s + 0.744·65-s − 0.855·67-s − 0.712·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65480464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65480464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.600571446\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.600571446\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 17 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 + T + p T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - T + 7 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 - 6 T + 22 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 6 T + 34 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 + 5 T + 65 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 2 T + 62 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 3 T + 55 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 11 T + 113 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 2 T + 82 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 11 T + 133 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 5 T + 99 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 7 T + 143 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 6 T + 34 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 5 T + 123 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 18 T + 226 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 + 26 T + 334 T^{2} + 26 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 9 T + 211 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
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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
−7.946423738174697943832676474266, −7.55535209010803704918505924384, −7.17818593019336037934643177946, −7.16132020828366039380091550104, −6.32860878264655289987567070582, −6.13042040401507877114595094373, −6.00728916213457089992248245994, −5.72698731312553703426512554854, −5.33554989996447177568795594665, −5.01585384101596546287586421020, −4.45661995383296930312310613812, −3.98368099772383706997406110071, −3.64207892064731498697424830735, −3.44352690751713961120840084827, −2.73973253372650843360912489036, −2.70321100912511630441997962618, −1.85726768790780131108194570529, −1.51439972219322667315559600030, −0.934653797610066712946965104778, −0.36289959752324163265750290482,
0.36289959752324163265750290482, 0.934653797610066712946965104778, 1.51439972219322667315559600030, 1.85726768790780131108194570529, 2.70321100912511630441997962618, 2.73973253372650843360912489036, 3.44352690751713961120840084827, 3.64207892064731498697424830735, 3.98368099772383706997406110071, 4.45661995383296930312310613812, 5.01585384101596546287586421020, 5.33554989996447177568795594665, 5.72698731312553703426512554854, 6.00728916213457089992248245994, 6.13042040401507877114595094373, 6.32860878264655289987567070582, 7.16132020828366039380091550104, 7.17818593019336037934643177946, 7.55535209010803704918505924384, 7.946423738174697943832676474266
