L(s) = 1 | + 5-s − 2·9-s + 6·13-s − 8·17-s − 2·25-s − 2·29-s − 14·37-s − 3·41-s − 2·45-s + 8·49-s + 8·53-s + 7·61-s + 6·65-s + 4·73-s − 5·81-s − 8·85-s + 2·89-s − 14·97-s + 20·101-s + 10·109-s + 8·113-s − 12·117-s − 8·121-s − 10·125-s + 127-s + 131-s + 137-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 2/3·9-s + 1.66·13-s − 1.94·17-s − 2/5·25-s − 0.371·29-s − 2.30·37-s − 0.468·41-s − 0.298·45-s + 8/7·49-s + 1.09·53-s + 0.896·61-s + 0.744·65-s + 0.468·73-s − 5/9·81-s − 0.867·85-s + 0.211·89-s − 1.42·97-s + 1.99·101-s + 0.957·109-s + 0.752·113-s − 1.10·117-s − 0.727·121-s − 0.894·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800320 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800320 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | | \( 1 \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 2 T + p T^{2} ) \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 2 T + p T^{2} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 6 T + p T^{2} ) \) |
good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 100 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 52 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 48 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
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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
−8.247658000180988947890253514716, −7.48168001799794547999928985232, −7.11914929389918597980963935584, −6.54019652352669494820665245836, −6.31226569694325206048189619321, −5.79945991911009684576091154250, −5.32394306642124884865393324008, −4.93240178143531949091411594764, −4.11871130856689840336310358761, −3.80130586509208040180634531543, −3.29104930477641380857456310406, −2.42299562930243412872376117621, −2.04867386587910704648619430628, −1.23139974976646853808284811930, 0,
1.23139974976646853808284811930, 2.04867386587910704648619430628, 2.42299562930243412872376117621, 3.29104930477641380857456310406, 3.80130586509208040180634531543, 4.11871130856689840336310358761, 4.93240178143531949091411594764, 5.32394306642124884865393324008, 5.79945991911009684576091154250, 6.31226569694325206048189619321, 6.54019652352669494820665245836, 7.11914929389918597980963935584, 7.48168001799794547999928985232, 8.247658000180988947890253514716