| L(s) = 1 | − 3·3-s − 3·5-s + 2·7-s + 4·9-s + 2·11-s − 10·13-s + 9·15-s − 2·17-s − 3·19-s − 6·21-s − 3·23-s − 6·27-s + 17·29-s + 11·31-s − 6·33-s − 6·35-s + 6·37-s + 30·39-s − 2·41-s − 16·43-s − 12·45-s − 18·47-s + 3·49-s + 6·51-s − 9·53-s − 6·55-s + 9·57-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 1.34·5-s + 0.755·7-s + 4/3·9-s + 0.603·11-s − 2.77·13-s + 2.32·15-s − 0.485·17-s − 0.688·19-s − 1.30·21-s − 0.625·23-s − 1.15·27-s + 3.15·29-s + 1.97·31-s − 1.04·33-s − 1.01·35-s + 0.986·37-s + 4.80·39-s − 0.312·41-s − 2.43·43-s − 1.78·45-s − 2.62·47-s + 3/7·49-s + 0.840·51-s − 1.23·53-s − 0.809·55-s + 1.19·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1317904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1317904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 41 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_4$ | \( 1 + p T + 5 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 3 T + 9 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 + 2 T - p T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 3 T + 11 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 3 T + 45 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 17 T + 127 T^{2} - 17 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 11 T + 63 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 6 T + 70 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 16 T + 137 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 9 T + 123 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - T + 89 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 6 T + 79 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 7 T + 117 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 10 T + 115 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 + 26 T + 314 T^{2} + 26 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 49 T^{2} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - T + 97 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 7 T + 47 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
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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
−9.961414740221143141082937553408, −9.359829169732999118224323952834, −8.458997331607138900203396001902, −8.282082728411612588929065157092, −8.024447531732714624305872337895, −7.48978747266291864979269984048, −6.98833163798159531286378604341, −6.66691879071017716711945135472, −6.25276846102323624839260353989, −5.91878841737183448085647024331, −4.95062669236222872955198003075, −4.84879565949978601910816944378, −4.53582088453010019548437907418, −4.36822042737295187210412247150, −3.34860006362171274484112762775, −2.79560697896910721361933140368, −2.07949825066371185577319425321, −1.22097015233537491478469959868, 0, 0,
1.22097015233537491478469959868, 2.07949825066371185577319425321, 2.79560697896910721361933140368, 3.34860006362171274484112762775, 4.36822042737295187210412247150, 4.53582088453010019548437907418, 4.84879565949978601910816944378, 4.95062669236222872955198003075, 5.91878841737183448085647024331, 6.25276846102323624839260353989, 6.66691879071017716711945135472, 6.98833163798159531286378604341, 7.48978747266291864979269984048, 8.024447531732714624305872337895, 8.282082728411612588929065157092, 8.458997331607138900203396001902, 9.359829169732999118224323952834, 9.961414740221143141082937553408
