| L(s) = 1 | − 3-s − 3·5-s + 2·7-s − 9-s + 2·13-s + 3·15-s + 4·17-s + 2·19-s − 2·21-s − 7·23-s + 25-s + 4·29-s − 17·31-s − 6·35-s + 5·37-s − 2·39-s + 20·41-s − 8·43-s + 3·45-s + 4·47-s + 3·49-s − 4·51-s + 8·53-s − 2·57-s − 15·59-s − 8·61-s − 2·63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.34·5-s + 0.755·7-s − 1/3·9-s + 0.554·13-s + 0.774·15-s + 0.970·17-s + 0.458·19-s − 0.436·21-s − 1.45·23-s + 1/5·25-s + 0.742·29-s − 3.05·31-s − 1.01·35-s + 0.821·37-s − 0.320·39-s + 3.12·41-s − 1.21·43-s + 0.447·45-s + 0.583·47-s + 3/7·49-s − 0.560·51-s + 1.09·53-s − 0.264·57-s − 1.95·59-s − 1.02·61-s − 0.251·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45914176 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45914176 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_4$ | \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 7 T + 54 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 + 17 T + 130 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 5 T + 76 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 15 T + 170 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + T + 130 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 5 T + 110 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 8 T + 94 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 2 T + 142 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 + 21 T + 284 T^{2} + 21 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 13 T + 232 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
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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
−7.77267234439599828279592930901, −7.52807244477523977203161271699, −7.29786509197751487805626380862, −6.87206484717902914504319962563, −6.19944182302415655194744581296, −5.99982328875424387942571806626, −5.57960540307878158835134197716, −5.54285930231328525424556792476, −4.98929567408074377397827291788, −4.30888889305250728453149327499, −4.24815577570749729812860521751, −3.96871247875509430894966132480, −3.32702265487708601549182623848, −3.22644033963672142558675996812, −2.45615586923298913303358862890, −2.07656987191789601156764518747, −1.30942469510242811071717685274, −1.11032867739664032807415032881, 0, 0,
1.11032867739664032807415032881, 1.30942469510242811071717685274, 2.07656987191789601156764518747, 2.45615586923298913303358862890, 3.22644033963672142558675996812, 3.32702265487708601549182623848, 3.96871247875509430894966132480, 4.24815577570749729812860521751, 4.30888889305250728453149327499, 4.98929567408074377397827291788, 5.54285930231328525424556792476, 5.57960540307878158835134197716, 5.99982328875424387942571806626, 6.19944182302415655194744581296, 6.87206484717902914504319962563, 7.29786509197751487805626380862, 7.52807244477523977203161271699, 7.77267234439599828279592930901
