L(s) = 1 | + 5-s + 5·9-s + 12·13-s + 5·17-s − 2·25-s + 4·29-s − 21·37-s − 41-s + 5·45-s + 10·49-s + 53-s + 2·61-s + 12·65-s − 4·73-s + 16·81-s + 5·85-s − 16·89-s − 7·97-s − 28·101-s + 7·109-s + 11·113-s + 60·117-s − 5·121-s − 10·125-s + 127-s + 131-s + 137-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 5/3·9-s + 3.32·13-s + 1.21·17-s − 2/5·25-s + 0.742·29-s − 3.45·37-s − 0.156·41-s + 0.745·45-s + 10/7·49-s + 0.137·53-s + 0.256·61-s + 1.48·65-s − 0.468·73-s + 16/9·81-s + 0.542·85-s − 1.69·89-s − 0.710·97-s − 2.78·101-s + 0.670·109-s + 1.03·113-s + 5.54·117-s − 0.454·121-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
\(L(1)\) |
\(\approx\) |
\(3.476411754\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.476411754\) |
\(L(\frac{3}{2})\) |
|
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$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 3 T + p T^{2} ) \) |
good | 3 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 5 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 41 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 83 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 56 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 64 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 112 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−8.288559563281819420350494292332, −7.942313961823255569318744323020, −7.17887758609772624441644062622, −6.88427827452403242273004071673, −6.58018193046949069737628023531, −5.91441632932643969745087073460, −5.59710797076793673100512138178, −5.23349678651743309954780579447, −4.28108882289940864842916454478, −4.07172866143636139222097398660, −3.45863745802712588196399639346, −3.20707633685834028559904709709, −1.99302602612246717545843835322, −1.40628897199210914480781914950, −1.11523311835702496158253562486,
1.11523311835702496158253562486, 1.40628897199210914480781914950, 1.99302602612246717545843835322, 3.20707633685834028559904709709, 3.45863745802712588196399639346, 4.07172866143636139222097398660, 4.28108882289940864842916454478, 5.23349678651743309954780579447, 5.59710797076793673100512138178, 5.91441632932643969745087073460, 6.58018193046949069737628023531, 6.88427827452403242273004071673, 7.17887758609772624441644062622, 7.942313961823255569318744323020, 8.288559563281819420350494292332