| L(s) = 1 | − 2·2-s + 3·4-s + 2·5-s − 4·8-s − 4·10-s − 2·11-s + 8·13-s + 5·16-s − 10·17-s + 4·19-s + 6·20-s + 4·22-s − 2·23-s − 5·25-s − 16·26-s − 4·31-s − 6·32-s + 20·34-s + 8·37-s − 8·38-s − 8·40-s − 2·41-s − 6·44-s + 4·46-s − 10·47-s + 10·50-s + 24·52-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s + 0.894·5-s − 1.41·8-s − 1.26·10-s − 0.603·11-s + 2.21·13-s + 5/4·16-s − 2.42·17-s + 0.917·19-s + 1.34·20-s + 0.852·22-s − 0.417·23-s − 25-s − 3.13·26-s − 0.718·31-s − 1.06·32-s + 3.42·34-s + 1.31·37-s − 1.29·38-s − 1.26·40-s − 0.312·41-s − 0.904·44-s + 0.589·46-s − 1.45·47-s + 1.41·50-s + 3.32·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 94128804 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 94128804 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.042714952\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.042714952\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - 2 T + 9 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 8 T + 34 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 10 T + 3 p T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 4 T + 34 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2 T + 29 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 31 | $C_4$ | \( 1 + 4 T - 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 8 T + 58 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 2 T + 51 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 10 T + 101 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 16 T + 162 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 4 T + 90 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 6 T + 129 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 10 T + 87 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 4 T + 142 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 18 T + 221 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 14 T + 207 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 8 T + 122 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 26 T + 355 T^{2} - 26 p T^{3} + p^{2} T^{4} \) |
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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
−7.75980098692441809409520256784, −7.71830294762776455579518831457, −7.19430754090099976546372763906, −6.90751428936773336800017687048, −6.39150438288856625726644122144, −6.28709005722480928707401040083, −5.92361095041523925319748316832, −5.76163771037126274965584181007, −5.20310546058544342727784695603, −4.86308031073735847406379559640, −4.21959379999979578918023023464, −4.02711169087787486626797627670, −3.38484590066029595368408930185, −3.26188180739200364696851324383, −2.51978016143408214864950207166, −2.18433459745898235871165962937, −1.83639033109348372347915691453, −1.62165681714596334888669739203, −0.71143360553625011797206186824, −0.56690169144027953126232582918,
0.56690169144027953126232582918, 0.71143360553625011797206186824, 1.62165681714596334888669739203, 1.83639033109348372347915691453, 2.18433459745898235871165962937, 2.51978016143408214864950207166, 3.26188180739200364696851324383, 3.38484590066029595368408930185, 4.02711169087787486626797627670, 4.21959379999979578918023023464, 4.86308031073735847406379559640, 5.20310546058544342727784695603, 5.76163771037126274965584181007, 5.92361095041523925319748316832, 6.28709005722480928707401040083, 6.39150438288856625726644122144, 6.90751428936773336800017687048, 7.19430754090099976546372763906, 7.71830294762776455579518831457, 7.75980098692441809409520256784
