| L(s) = 1 | − 2-s − 4-s + 3·8-s − 6·9-s − 4·13-s − 16-s − 12·17-s + 6·18-s + 4·26-s + 12·29-s − 5·32-s + 12·34-s + 6·36-s + 4·37-s + 4·41-s − 14·49-s + 4·52-s + 4·53-s − 12·58-s − 20·61-s + 7·64-s + 12·68-s − 18·72-s − 28·73-s − 4·74-s + 27·81-s − 4·82-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 1.06·8-s − 2·9-s − 1.10·13-s − 1/4·16-s − 2.91·17-s + 1.41·18-s + 0.784·26-s + 2.22·29-s − 0.883·32-s + 2.05·34-s + 36-s + 0.657·37-s + 0.624·41-s − 2·49-s + 0.554·52-s + 0.549·53-s − 1.57·58-s − 2.56·61-s + 7/8·64-s + 1.45·68-s − 2.12·72-s − 3.27·73-s − 0.464·74-s + 3·81-s − 0.441·82-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1210000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1210000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 5 | | \( 1 \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.82885193609770565758085877155, −7.25409359989160561546791965971, −6.54918508631964327843106616026, −6.48067063437882788912913410261, −5.91522945024739734859341901763, −5.23353820210759730524176244525, −4.84786612010371112418366573978, −4.45768242969291673579097656708, −4.11816586584967359873614255585, −3.07429403089419103076246288907, −2.69402268918089176010114790698, −2.31804956482818418780108048277, −1.35944850033720564570853590086, 0, 0,
1.35944850033720564570853590086, 2.31804956482818418780108048277, 2.69402268918089176010114790698, 3.07429403089419103076246288907, 4.11816586584967359873614255585, 4.45768242969291673579097656708, 4.84786612010371112418366573978, 5.23353820210759730524176244525, 5.91522945024739734859341901763, 6.48067063437882788912913410261, 6.54918508631964327843106616026, 7.25409359989160561546791965971, 7.82885193609770565758085877155
