| L(s) = 1 | − 3·2-s + 4·4-s − 4·7-s − 3·8-s − 9-s + 2·11-s + 4·13-s + 12·14-s + 3·16-s − 3·17-s + 3·18-s − 6·22-s + 2·23-s − 12·26-s − 16·28-s − 12·29-s − 3·31-s − 6·32-s + 9·34-s − 4·36-s − 15·37-s + 15·41-s + 43-s + 8·44-s − 6·46-s − 13·47-s + 3·49-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 2·4-s − 1.51·7-s − 1.06·8-s − 1/3·9-s + 0.603·11-s + 1.10·13-s + 3.20·14-s + 3/4·16-s − 0.727·17-s + 0.707·18-s − 1.27·22-s + 0.417·23-s − 2.35·26-s − 3.02·28-s − 2.22·29-s − 0.538·31-s − 1.06·32-s + 1.54·34-s − 2/3·36-s − 2.46·37-s + 2.34·41-s + 0.152·43-s + 1.20·44-s − 0.884·46-s − 1.89·47-s + 3/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81450625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81450625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 19 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 4 T + 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2 T + 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 + 3 T + 25 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2 T + 42 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 + 3 T + 53 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 15 T + 119 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 15 T + 137 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - T - 15 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 13 T + 135 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 15 T + 161 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 10 T + 138 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 9 T + 41 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4 T + 93 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 18 T + 203 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 8 T + 117 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 + 9 T + 155 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 6 T + 142 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 17 T + 265 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
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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
−7.49194715309917211490484652372, −7.35531475183348738192944558297, −6.99773052779431686257558485758, −6.71619083831045008883991278067, −6.38023084479049233426945192922, −5.93225916557088466091950702047, −5.75132689134770131040437761502, −5.28038611522135665237498538858, −4.97546596505039016273070379263, −4.05430363955048767070631281066, −3.93287738711975059855197663193, −3.58810936415103998876882393207, −3.25956834176814009342240549881, −2.74520327274596273835446013246, −2.10145415396515924583231871878, −1.90307919478012105254631857328, −1.17378629497462513275111252825, −0.845619292405495196405407857337, 0, 0,
0.845619292405495196405407857337, 1.17378629497462513275111252825, 1.90307919478012105254631857328, 2.10145415396515924583231871878, 2.74520327274596273835446013246, 3.25956834176814009342240549881, 3.58810936415103998876882393207, 3.93287738711975059855197663193, 4.05430363955048767070631281066, 4.97546596505039016273070379263, 5.28038611522135665237498538858, 5.75132689134770131040437761502, 5.93225916557088466091950702047, 6.38023084479049233426945192922, 6.71619083831045008883991278067, 6.99773052779431686257558485758, 7.35531475183348738192944558297, 7.49194715309917211490484652372
