| L(s) = 1 | + 2·3-s + 4-s − 2·5-s + 3·9-s + 2·12-s − 4·15-s + 16-s − 2·20-s + 3·25-s + 4·27-s − 8·31-s + 3·36-s + 4·37-s − 6·45-s − 24·47-s + 2·48-s + 49-s + 12·53-s − 24·59-s − 4·60-s + 64-s + 16·67-s + 6·75-s − 2·80-s + 5·81-s + 12·89-s − 16·93-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/2·4-s − 0.894·5-s + 9-s + 0.577·12-s − 1.03·15-s + 1/4·16-s − 0.447·20-s + 3/5·25-s + 0.769·27-s − 1.43·31-s + 1/2·36-s + 0.657·37-s − 0.894·45-s − 3.50·47-s + 0.288·48-s + 1/7·49-s + 1.64·53-s − 3.12·59-s − 0.516·60-s + 1/8·64-s + 1.95·67-s + 0.692·75-s − 0.223·80-s + 5/9·81-s + 1.27·89-s − 1.65·93-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5336100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5336100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 11 | $C_2$ | \( 1 + p T^{2} \) |
| good | 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
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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
−7.27965583167809929942184453108, −6.82344471735118290623692578055, −6.25947002831397401127236915833, −6.14620005069006563789988203374, −5.32658287662016836935900647118, −4.86401254971600220769159288355, −4.64849576847169223614699526425, −3.83361606782573647022022560269, −3.71822253537069739639390686069, −3.20889430817744334094426640318, −2.84067104557978538300695538211, −2.16105217917237072156284139202, −1.76843065738470470879189593099, −1.06078487395365503971461934513, 0,
1.06078487395365503971461934513, 1.76843065738470470879189593099, 2.16105217917237072156284139202, 2.84067104557978538300695538211, 3.20889430817744334094426640318, 3.71822253537069739639390686069, 3.83361606782573647022022560269, 4.64849576847169223614699526425, 4.86401254971600220769159288355, 5.32658287662016836935900647118, 6.14620005069006563789988203374, 6.25947002831397401127236915833, 6.82344471735118290623692578055, 7.27965583167809929942184453108
