| L(s) = 1 | − 2·4-s + 4·7-s + 9·13-s + 5·25-s − 8·28-s + 18·31-s − 22·37-s + 8·43-s + 9·49-s − 18·52-s + 27·61-s + 8·64-s − 5·67-s − 17·79-s + 36·91-s − 9·97-s − 10·100-s − 27·103-s − 4·109-s − 11·121-s − 36·124-s + 127-s + 131-s + 137-s + 139-s + 44·148-s + 149-s + ⋯ |
| L(s) = 1 | − 4-s + 1.51·7-s + 2.49·13-s + 25-s − 1.51·28-s + 3.23·31-s − 3.61·37-s + 1.21·43-s + 9/7·49-s − 2.49·52-s + 3.45·61-s + 64-s − 0.610·67-s − 1.91·79-s + 3.77·91-s − 0.913·97-s − 100-s − 2.66·103-s − 0.383·109-s − 121-s − 3.23·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 3.61·148-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 321489 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 321489 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.043182272\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.043182272\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 13 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−10.94568028476647276280431199318, −10.66174453442076490107849856037, −10.02650695254519338829399270359, −9.792635039232500839134158563523, −8.839544318817663542280157459757, −8.631174265785056806418123754405, −8.421112205860520235729250153707, −8.320186933738542966062270145530, −7.43428625947642816894114276850, −6.86539205183067490788031982889, −6.47014582724734271648677274731, −5.85306585121396278632541510528, −5.13949734779561967109687541268, −5.08791430922429896050538253908, −4.13917340566943113438696120691, −4.12369611000396327449666375077, −3.31771300176373423347907024641, −2.49772205195111361636446926241, −1.46878458416884065570138314693, −0.984617890730932138821312825415,
0.984617890730932138821312825415, 1.46878458416884065570138314693, 2.49772205195111361636446926241, 3.31771300176373423347907024641, 4.12369611000396327449666375077, 4.13917340566943113438696120691, 5.08791430922429896050538253908, 5.13949734779561967109687541268, 5.85306585121396278632541510528, 6.47014582724734271648677274731, 6.86539205183067490788031982889, 7.43428625947642816894114276850, 8.320186933738542966062270145530, 8.421112205860520235729250153707, 8.631174265785056806418123754405, 8.839544318817663542280157459757, 9.792635039232500839134158563523, 10.02650695254519338829399270359, 10.66174453442076490107849856037, 10.94568028476647276280431199318
