| L(s) = 1 | − 9-s − 4·17-s − 2·25-s + 6·41-s − 4·49-s − 2·73-s + 81-s + 6·97-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 4·153-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | − 9-s − 4·17-s − 2·25-s + 6·41-s − 4·49-s − 2·73-s + 81-s + 6·97-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 4·153-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5113773246\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5113773246\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 19 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| good | 3 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 5 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 11 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 - T + T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 59 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 - T + T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.15159434707151250042789265430, −6.97711602030787978597865787828, −6.64648105408904582220075486021, −6.33257708674911452047802337702, −6.24973198387071939405427140802, −6.10045168952924732326500265181, −6.07511311539990027229873017074, −5.63838339759436784657694854294, −5.53776359687503644312219280328, −5.08077694003217113156156636332, −4.80308151849610231064714743141, −4.63205338091576310812121474394, −4.34459692288072221695634179016, −4.27618592708093664970233098100, −4.14816371993926317542914870285, −3.77572869709346389763175668568, −3.20312060153324186331548047291, −3.14486198115345709746484606932, −3.00488165892666332850729556221, −2.33145767731444444691075459073, −2.21709606127198372665030073163, −2.09417791663724743545999534168, −1.92954769839779621893925985129, −1.16938147683957324175620717374, −0.49780959635767278189143083260,
0.49780959635767278189143083260, 1.16938147683957324175620717374, 1.92954769839779621893925985129, 2.09417791663724743545999534168, 2.21709606127198372665030073163, 2.33145767731444444691075459073, 3.00488165892666332850729556221, 3.14486198115345709746484606932, 3.20312060153324186331548047291, 3.77572869709346389763175668568, 4.14816371993926317542914870285, 4.27618592708093664970233098100, 4.34459692288072221695634179016, 4.63205338091576310812121474394, 4.80308151849610231064714743141, 5.08077694003217113156156636332, 5.53776359687503644312219280328, 5.63838339759436784657694854294, 6.07511311539990027229873017074, 6.10045168952924732326500265181, 6.24973198387071939405427140802, 6.33257708674911452047802337702, 6.64648105408904582220075486021, 6.97711602030787978597865787828, 7.15159434707151250042789265430
