L(s) = 1 | + 2·2-s + 4-s − 2·5-s + 4·7-s − 4·10-s − 4·11-s − 2·13-s + 8·14-s + 16-s + 4·17-s + 4·19-s − 2·20-s − 8·22-s + 3·25-s − 4·26-s + 4·28-s + 12·31-s − 2·32-s + 8·34-s − 8·35-s + 8·38-s + 12·41-s − 8·43-s − 4·44-s + 4·47-s + 6·49-s + 6·50-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 1/2·4-s − 0.894·5-s + 1.51·7-s − 1.26·10-s − 1.20·11-s − 0.554·13-s + 2.13·14-s + 1/4·16-s + 0.970·17-s + 0.917·19-s − 0.447·20-s − 1.70·22-s + 3/5·25-s − 0.784·26-s + 0.755·28-s + 2.15·31-s − 0.353·32-s + 1.37·34-s − 1.35·35-s + 1.29·38-s + 1.87·41-s − 1.21·43-s − 0.603·44-s + 0.583·47-s + 6/7·49-s + 0.848·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 342225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.441391519\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.441391519\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
good | 2 | $D_{4}$ | \( 1 - p T + 3 T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 7 | $C_4$ | \( 1 - 4 T + 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 4 T + 24 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 4 T + 40 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 44 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 12 T + 80 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 12 T + 110 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 8 T + 52 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 4 T + 90 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 12 T + 70 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 12 T + 136 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 + 4 T + 48 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 12 T + 194 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 + 4 T + 166 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
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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
−11.02327904860097487853365600695, −10.49217856516854857244707613659, −10.36110934961180151067116332977, −9.714286165109304329331025621075, −9.094636004158736703950196942447, −8.601144137217098123137355612783, −7.931217069701776126282701574024, −7.82748707143792572786159821091, −7.52806058553861407863894042151, −7.00274043662204370004099982388, −6.06463888489816548881869814765, −5.70839080855422604232435173677, −5.06077252307295835194499274723, −4.89361651758623846143271627664, −4.42392998264889679461814329749, −4.08089846690771216958976744620, −3.07921870702749460485365586304, −2.97450618779523508374006928154, −1.91204956050814003374058696388, −0.897256041534199990641944032128,
0.897256041534199990641944032128, 1.91204956050814003374058696388, 2.97450618779523508374006928154, 3.07921870702749460485365586304, 4.08089846690771216958976744620, 4.42392998264889679461814329749, 4.89361651758623846143271627664, 5.06077252307295835194499274723, 5.70839080855422604232435173677, 6.06463888489816548881869814765, 7.00274043662204370004099982388, 7.52806058553861407863894042151, 7.82748707143792572786159821091, 7.931217069701776126282701574024, 8.601144137217098123137355612783, 9.094636004158736703950196942447, 9.714286165109304329331025621075, 10.36110934961180151067116332977, 10.49217856516854857244707613659, 11.02327904860097487853365600695