| L(s) = 1 | + 2-s − 2·4-s − 2·7-s − 3·8-s − 9-s − 6·11-s − 6·13-s − 2·14-s + 16-s − 6·17-s − 18-s − 4·19-s − 6·22-s − 2·23-s − 6·26-s + 4·28-s − 6·29-s + 2·32-s − 6·34-s + 2·36-s − 2·37-s − 4·38-s + 2·41-s + 12·44-s − 2·46-s − 6·49-s + 12·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 4-s − 0.755·7-s − 1.06·8-s − 1/3·9-s − 1.80·11-s − 1.66·13-s − 0.534·14-s + 1/4·16-s − 1.45·17-s − 0.235·18-s − 0.917·19-s − 1.27·22-s − 0.417·23-s − 1.17·26-s + 0.755·28-s − 1.11·29-s + 0.353·32-s − 1.02·34-s + 1/3·36-s − 0.328·37-s − 0.648·38-s + 0.312·41-s + 1.80·44-s − 0.294·46-s − 6/7·49-s + 1.66·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 330625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 330625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 23 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_4$ | \( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 + 6 T + 38 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 17 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 2 T + 70 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 2 T + 63 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 89 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 4 T + 102 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 10 T + 154 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 20 T + 237 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 22 T + 247 T^{2} + 22 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 4 T + 82 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 22 T + 282 T^{2} - 22 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_4$ | \( 1 + 12 T + 194 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 22 T + 270 T^{2} + 22 p T^{3} + p^{2} T^{4} \) |
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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
−10.76009312718436085231162643313, −9.867412352734324197405564353661, −9.547864028753442484523812998478, −9.513513290924372619826046740794, −8.526534161097957062624098187443, −8.485422600718401530227578063137, −7.962069160367340893428268238224, −7.16107845661655673232901680953, −7.03378099331315442277711605203, −6.32034116706534023990351271627, −5.59218035581207484624801786429, −5.41226095038832743760400764873, −4.85469906029089226711569754979, −4.42591766911930171458881528185, −3.93241212986199430247095182147, −3.25780320302808949390923485814, −2.38221053538701941109925317217, −2.32708019964692189665792439050, 0, 0,
2.32708019964692189665792439050, 2.38221053538701941109925317217, 3.25780320302808949390923485814, 3.93241212986199430247095182147, 4.42591766911930171458881528185, 4.85469906029089226711569754979, 5.41226095038832743760400764873, 5.59218035581207484624801786429, 6.32034116706534023990351271627, 7.03378099331315442277711605203, 7.16107845661655673232901680953, 7.962069160367340893428268238224, 8.485422600718401530227578063137, 8.526534161097957062624098187443, 9.513513290924372619826046740794, 9.547864028753442484523812998478, 9.867412352734324197405564353661, 10.76009312718436085231162643313
