| L(s) = 1 | − 3·3-s − 5·5-s − 5·7-s + 3·9-s − 2·13-s + 15·15-s − 8·17-s − 2·19-s + 15·21-s − 23-s + 10·25-s + 3·29-s + 3·31-s + 25·35-s − 37-s + 6·39-s − 41-s + 2·43-s − 15·45-s + 47-s + 8·49-s + 24·51-s − 7·53-s + 6·57-s + 59-s − 6·61-s − 15·63-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 2.23·5-s − 1.88·7-s + 9-s − 0.554·13-s + 3.87·15-s − 1.94·17-s − 0.458·19-s + 3.27·21-s − 0.208·23-s + 2·25-s + 0.557·29-s + 0.538·31-s + 4.22·35-s − 0.164·37-s + 0.960·39-s − 0.156·41-s + 0.304·43-s − 2.23·45-s + 0.145·47-s + 8/7·49-s + 3.36·51-s − 0.961·53-s + 0.794·57-s + 0.130·59-s − 0.768·61-s − 1.88·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26384 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26384 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 7 T + p T^{2} ) \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 12 T + p T^{2} ) \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_4$ | \( 1 + p T + 3 p T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 5 T + 17 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 20 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + T - 14 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 3 T + p T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 3 T - 18 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + T + 27 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + T + 20 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 2 T - 50 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - T + 43 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 7 T + 100 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - T + 72 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 6 T + 24 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 12 T + 152 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 4 T - 14 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 16 T + 200 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_4$ | \( 1 + 4 T + 58 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 6 T + 88 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 9 T + 47 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
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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
−15.9585300603, −15.6172039990, −15.2785130970, −14.8269477845, −13.9128197342, −13.3091162839, −12.8759062541, −12.3407219062, −12.0848789043, −11.6576695087, −11.2159667822, −10.8717532608, −10.3034331237, −9.70977773476, −9.01166944544, −8.49144212845, −7.77944752612, −7.17004322956, −6.66070953100, −6.29713053712, −5.70184322571, −4.60360722797, −4.40001721139, −3.56799649235, −2.77973599304, 0, 0,
2.77973599304, 3.56799649235, 4.40001721139, 4.60360722797, 5.70184322571, 6.29713053712, 6.66070953100, 7.17004322956, 7.77944752612, 8.49144212845, 9.01166944544, 9.70977773476, 10.3034331237, 10.8717532608, 11.2159667822, 11.6576695087, 12.0848789043, 12.3407219062, 12.8759062541, 13.3091162839, 13.9128197342, 14.8269477845, 15.2785130970, 15.6172039990, 15.9585300603
