L(s) = 1 | + 2·5-s + 2·7-s + 4·11-s − 8·13-s − 6·17-s + 2·19-s − 12·23-s + 6·29-s − 8·31-s + 4·35-s + 4·41-s − 4·43-s + 10·47-s + 3·49-s − 2·53-s + 8·55-s + 8·59-s + 16·61-s − 16·65-s + 8·67-s + 6·71-s − 8·73-s + 8·77-s + 12·79-s + 18·83-s − 12·85-s + 12·89-s + ⋯ |
L(s) = 1 | + 0.894·5-s + 0.755·7-s + 1.20·11-s − 2.21·13-s − 1.45·17-s + 0.458·19-s − 2.50·23-s + 1.11·29-s − 1.43·31-s + 0.676·35-s + 0.624·41-s − 0.609·43-s + 1.45·47-s + 3/7·49-s − 0.274·53-s + 1.07·55-s + 1.04·59-s + 2.04·61-s − 1.98·65-s + 0.977·67-s + 0.712·71-s − 0.936·73-s + 0.911·77-s + 1.35·79-s + 1.97·83-s − 1.30·85-s + 1.27·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91699776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91699776 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.477244639\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.477244639\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
good | 5 | $D_{4}$ | \( 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 + 6 T + 36 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 - 6 T + 60 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 8 T + 50 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 4 T - 26 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 - 10 T + 112 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 2 T + 44 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 8 T + 22 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 16 T + 158 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 8 T + 38 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 6 T + 144 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 8 T + 134 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 12 T + 166 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 18 T + 184 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 - 12 T + 118 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
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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
−7.74697340580436266784098596421, −7.42123977774045377443152477608, −7.34251128776754266461011117422, −6.69716885087954619688666436482, −6.46805534521897192994987711042, −6.35763427696435639169456475604, −5.62987357478536791566424374195, −5.58419501201243826331107918807, −5.05907130540039012820748234041, −4.82307016569430279930741481353, −4.37129029911857361233485314990, −4.10904536613812480863525333843, −3.69515925652149566266726443930, −3.32028826377658348341060409570, −2.44520358434183380005776173201, −2.30829165602951335449140978382, −1.93922235124099235517751480010, −1.88123004508026138895897625020, −0.860121756746707250230110630271, −0.48068620101203434500420582618,
0.48068620101203434500420582618, 0.860121756746707250230110630271, 1.88123004508026138895897625020, 1.93922235124099235517751480010, 2.30829165602951335449140978382, 2.44520358434183380005776173201, 3.32028826377658348341060409570, 3.69515925652149566266726443930, 4.10904536613812480863525333843, 4.37129029911857361233485314990, 4.82307016569430279930741481353, 5.05907130540039012820748234041, 5.58419501201243826331107918807, 5.62987357478536791566424374195, 6.35763427696435639169456475604, 6.46805534521897192994987711042, 6.69716885087954619688666436482, 7.34251128776754266461011117422, 7.42123977774045377443152477608, 7.74697340580436266784098596421