| L(s) = 1 | + 4·2-s − 6·3-s + 12·4-s − 5·5-s − 24·6-s + 32·8-s + 27·9-s − 20·10-s + 67·11-s − 72·12-s − 41·13-s + 30·15-s + 80·16-s + 92·17-s + 108·18-s − 43·19-s − 60·20-s + 268·22-s + 148·23-s − 192·24-s + 105·25-s − 164·26-s − 108·27-s + 77·29-s + 120·30-s + 520·31-s + 192·32-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s + 3/2·4-s − 0.447·5-s − 1.63·6-s + 1.41·8-s + 9-s − 0.632·10-s + 1.83·11-s − 1.73·12-s − 0.874·13-s + 0.516·15-s + 5/4·16-s + 1.31·17-s + 1.41·18-s − 0.519·19-s − 0.670·20-s + 2.59·22-s + 1.34·23-s − 1.63·24-s + 0.839·25-s − 1.23·26-s − 0.769·27-s + 0.493·29-s + 0.730·30-s + 3.01·31-s + 1.06·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 86436 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 86436 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.439015364\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.439015364\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 3 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 + p T - 16 p T^{2} + p^{4} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 67 T + 3448 T^{2} - 67 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 41 T + 4478 T^{2} + 41 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 92 T + 386 p T^{2} - 92 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 43 T + 11154 T^{2} + 43 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 148 T + 24430 T^{2} - 148 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 77 T + 9574 T^{2} - 77 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 520 T + 125837 T^{2} - 520 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 7 T + 74082 T^{2} - 7 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 426 T + 171106 T^{2} - 426 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 107 T + 86220 T^{2} + 107 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 576 T + 242170 T^{2} + 576 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 243 T + 309490 T^{2} + 243 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 7 T + 200614 T^{2} - 7 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 224 T + 461126 T^{2} + 224 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 687 T + 678832 T^{2} - 687 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 472 T + 637018 T^{2} - 472 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 921 T + 578188 T^{2} - 921 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 526 T + 1033727 T^{2} + 526 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 221 T + 945628 T^{2} - 221 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 774 T + 966562 T^{2} + 774 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 1953 T + 2366992 T^{2} + 1953 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−11.54258345595655558889648991570, −11.50986501438528704966199010543, −10.87464253335143202706370729904, −10.41302766319001763613069802999, −9.696774776387679640969508786972, −9.592182095334407446812386647386, −8.618723815131448415811487056152, −8.006582193505075356085977511652, −7.51267993705453352825755030030, −6.72787293331951037098206427720, −6.49506659722780793942150125158, −6.34258580245333526702382353627, −5.27006793009996591775333387715, −5.09068099252254613152878113813, −4.38606685487803679683654327439, −4.09168588254448070600844980953, −3.22026884602611775519971206382, −2.65637026085848853284190149346, −1.33920226302345996389793189319, −0.868904609939446211252465652118,
0.868904609939446211252465652118, 1.33920226302345996389793189319, 2.65637026085848853284190149346, 3.22026884602611775519971206382, 4.09168588254448070600844980953, 4.38606685487803679683654327439, 5.09068099252254613152878113813, 5.27006793009996591775333387715, 6.34258580245333526702382353627, 6.49506659722780793942150125158, 6.72787293331951037098206427720, 7.51267993705453352825755030030, 8.006582193505075356085977511652, 8.618723815131448415811487056152, 9.592182095334407446812386647386, 9.696774776387679640969508786972, 10.41302766319001763613069802999, 10.87464253335143202706370729904, 11.50986501438528704966199010543, 11.54258345595655558889648991570
