| L(s) = 1 | + 3-s + 2·7-s − 9-s + 8·11-s − 3·13-s − 2·17-s − 8·19-s + 2·21-s + 2·23-s − 13·29-s − 7·31-s + 8·33-s + 6·37-s − 3·39-s − 3·41-s − 10·43-s + 5·47-s + 6·49-s − 2·51-s − 8·53-s − 8·57-s − 4·59-s + 14·61-s − 2·63-s − 4·67-s + 2·69-s + 5·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.755·7-s − 1/3·9-s + 2.41·11-s − 0.832·13-s − 0.485·17-s − 1.83·19-s + 0.436·21-s + 0.417·23-s − 2.41·29-s − 1.25·31-s + 1.39·33-s + 0.986·37-s − 0.480·39-s − 0.468·41-s − 1.52·43-s + 0.729·47-s + 6/7·49-s − 0.280·51-s − 1.09·53-s − 1.05·57-s − 0.520·59-s + 1.79·61-s − 0.251·63-s − 0.488·67-s + 0.240·69-s + 0.593·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84640000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84640000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.474581575\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.474581575\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 23 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 2 T - 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 + 3 T + 24 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 2 T + 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 + 13 T + 96 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 7 T + 70 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 6 T + 66 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 3 T + 80 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 10 T + 94 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 5 T + 62 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 14 T + 154 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 5 T + 110 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 29 T + 352 T^{2} + 29 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 4 T + 94 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 - 16 T + 190 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
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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
−7.85134227635075918275125901416, −7.45480139763667597375579683086, −7.37344667194435769128953510571, −6.80969123593344564895570093493, −6.50449685657283727127598002076, −6.37136954112793551399311566078, −5.86026382933567027342875817545, −5.53620681204273916318659825016, −4.96884265983938944240165325104, −4.81415229100274953150224858338, −4.29880855890434524108311684873, −3.95184980699086291238857053161, −3.65544640242403380565184233164, −3.51302689287560635109911466893, −2.64079581489962468221237313991, −2.43493218215084872038371293093, −1.83946166624433489070707842368, −1.68042119468169628636225684191, −1.17558344795246918171055634894, −0.25303825931015543503927748560,
0.25303825931015543503927748560, 1.17558344795246918171055634894, 1.68042119468169628636225684191, 1.83946166624433489070707842368, 2.43493218215084872038371293093, 2.64079581489962468221237313991, 3.51302689287560635109911466893, 3.65544640242403380565184233164, 3.95184980699086291238857053161, 4.29880855890434524108311684873, 4.81415229100274953150224858338, 4.96884265983938944240165325104, 5.53620681204273916318659825016, 5.86026382933567027342875817545, 6.37136954112793551399311566078, 6.50449685657283727127598002076, 6.80969123593344564895570093493, 7.37344667194435769128953510571, 7.45480139763667597375579683086, 7.85134227635075918275125901416
