| L(s) = 1 | + 2·3-s + 10·5-s − 16·7-s − 45·9-s + 90·11-s + 50·13-s + 20·15-s − 44·17-s − 108·19-s − 32·21-s − 28·23-s + 41·25-s − 134·27-s − 58·29-s + 66·31-s + 180·33-s − 160·35-s − 40·37-s + 100·39-s + 304·41-s + 130·43-s − 450·45-s − 514·47-s − 110·49-s − 88·51-s + 958·53-s + 900·55-s + ⋯ |
| L(s) = 1 | + 0.384·3-s + 0.894·5-s − 0.863·7-s − 5/3·9-s + 2.46·11-s + 1.06·13-s + 0.344·15-s − 0.627·17-s − 1.30·19-s − 0.332·21-s − 0.253·23-s + 0.327·25-s − 0.955·27-s − 0.371·29-s + 0.382·31-s + 0.949·33-s − 0.772·35-s − 0.177·37-s + 0.410·39-s + 1.15·41-s + 0.461·43-s − 1.49·45-s − 1.59·47-s − 0.320·49-s − 0.241·51-s + 2.48·53-s + 2.20·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3444736 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3444736 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.932426787\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.932426787\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 29 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - 2 T + 49 T^{2} - 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 2 p T + 59 T^{2} - 2 p^{4} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 16 T + 366 T^{2} + 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 90 T + 4633 T^{2} - 90 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 50 T + 4635 T^{2} - 50 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 44 T + 8774 T^{2} + 44 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 108 T + 16538 T^{2} + 108 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 28 T - 11974 T^{2} + 28 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 66 T - 10615 T^{2} - 66 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 40 T + 101610 T^{2} + 40 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 304 T + 122546 T^{2} - 304 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 130 T + 146385 T^{2} - 130 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 514 T + 214889 T^{2} + 514 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 958 T + 510971 T^{2} - 958 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 180 T + 43858 T^{2} - 180 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 1028 T + 717294 T^{2} + 1028 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 912 T + 807062 T^{2} - 912 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 796 T + 867290 T^{2} - 796 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 856 T + 775362 T^{2} + 856 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 318 T + 229433 T^{2} + 318 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 1828 T + 1970306 T^{2} - 1828 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 944 T + 1601618 T^{2} + 944 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 368 T + 1799202 T^{2} - 368 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.073610711614418324499108083097, −8.769479737751325647048835913566, −8.412495514299114916017531901731, −8.244756372905439831809539320694, −7.39741431500271757074391105961, −7.00556073560597870071801035131, −6.35809910755019332423044222800, −6.31928024839069932214993420526, −6.10804389688195153774177041288, −5.72742565749569061642830735776, −5.06715942564098621447472922715, −4.45592025332053249081284439596, −3.96169443977272417702898138007, −3.65226383290654908420890471325, −3.22588952422120851375556921165, −2.67342354536231267214817265604, −2.01156523622587117318376398948, −1.82960640149067584448321496804, −0.924704832233745873209505789863, −0.46397687471829912936621347208,
0.46397687471829912936621347208, 0.924704832233745873209505789863, 1.82960640149067584448321496804, 2.01156523622587117318376398948, 2.67342354536231267214817265604, 3.22588952422120851375556921165, 3.65226383290654908420890471325, 3.96169443977272417702898138007, 4.45592025332053249081284439596, 5.06715942564098621447472922715, 5.72742565749569061642830735776, 6.10804389688195153774177041288, 6.31928024839069932214993420526, 6.35809910755019332423044222800, 7.00556073560597870071801035131, 7.39741431500271757074391105961, 8.244756372905439831809539320694, 8.412495514299114916017531901731, 8.769479737751325647048835913566, 9.073610711614418324499108083097
