L(s) = 1 | − 2-s + 3-s + 6·5-s − 6-s + 7-s + 8-s − 6·10-s − 4·11-s + 7·13-s − 14-s + 6·15-s − 16-s + 5·17-s − 4·19-s + 21-s + 4·22-s + 4·23-s + 24-s + 17·25-s − 7·26-s − 27-s + 9·29-s − 6·30-s − 4·33-s − 5·34-s + 6·35-s − 7·37-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 2.68·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s − 1.89·10-s − 1.20·11-s + 1.94·13-s − 0.267·14-s + 1.54·15-s − 1/4·16-s + 1.21·17-s − 0.917·19-s + 0.218·21-s + 0.852·22-s + 0.834·23-s + 0.204·24-s + 17/5·25-s − 1.37·26-s − 0.192·27-s + 1.67·29-s − 1.09·30-s − 0.696·33-s − 0.857·34-s + 1.01·35-s − 1.15·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.772874620\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.772874620\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | $C_2$ | \( 1 - T + T^{2} \) |
| 13 | $C_2$ | \( 1 - 7 T + p T^{2} \) |
good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 4 T + 5 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 5 T + 8 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 4 T - 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 9 T + 52 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 7 T + 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 7 T + 8 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 8 T + 5 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 7 T - 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 12 T + 77 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 12 T + 73 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 18 T + 227 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
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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
−10.53343115532845636195354206352, −10.41030222580722124453489354693, −10.03784542072222927536970703625, −10.03446004932540000580444650919, −9.014479191310767591713347856317, −8.865907095545422819853026582045, −8.479032050064001711359886192138, −8.382321443687000781853379714171, −7.27980534391095969980245568447, −7.19512410512835180601112530032, −6.32151837875218101606599151865, −5.86304369308204668490255619496, −5.66005246864061023392537503436, −5.19928598269149437855311638599, −4.47035994590382656832482485186, −3.66893432987448413897029189361, −2.74247361109148928325391222304, −2.55518251274908605510094966011, −1.52907911793429289417876431628, −1.31391052449536356878880577333,
1.31391052449536356878880577333, 1.52907911793429289417876431628, 2.55518251274908605510094966011, 2.74247361109148928325391222304, 3.66893432987448413897029189361, 4.47035994590382656832482485186, 5.19928598269149437855311638599, 5.66005246864061023392537503436, 5.86304369308204668490255619496, 6.32151837875218101606599151865, 7.19512410512835180601112530032, 7.27980534391095969980245568447, 8.382321443687000781853379714171, 8.479032050064001711359886192138, 8.865907095545422819853026582045, 9.014479191310767591713347856317, 10.03446004932540000580444650919, 10.03784542072222927536970703625, 10.41030222580722124453489354693, 10.53343115532845636195354206352