| L(s) = 1 | + 3-s + 4-s + 7·7-s + 9-s + 12-s − 2·13-s + 16-s + 19-s + 7·21-s + 8·25-s + 27-s + 7·28-s + 10·31-s + 36-s − 2·37-s − 2·39-s − 11·43-s + 48-s + 25·49-s − 2·52-s + 57-s + 16·61-s + 7·63-s + 64-s − 5·67-s − 2·73-s + 8·75-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/2·4-s + 2.64·7-s + 1/3·9-s + 0.288·12-s − 0.554·13-s + 1/4·16-s + 0.229·19-s + 1.52·21-s + 8/5·25-s + 0.192·27-s + 1.32·28-s + 1.79·31-s + 1/6·36-s − 0.328·37-s − 0.320·39-s − 1.67·43-s + 0.144·48-s + 25/7·49-s − 0.277·52-s + 0.132·57-s + 2.04·61-s + 0.881·63-s + 1/8·64-s − 0.610·67-s − 0.234·73-s + 0.923·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 740772 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 740772 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.437258113\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.437258113\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 19 | $C_1$ | \( 1 - T \) |
| good | 5 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 64 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 53 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 119 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 128 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−8.198649989471353669686261061004, −8.112627193890949061808269468339, −7.34702615440567392596857320571, −7.11770500842890259526047007230, −6.73632075155858562443002999867, −5.97398016747507603353200514102, −5.39852558347754735104108081797, −4.94700711009357496864534098698, −4.64385789642771307621757561964, −4.26099745037811799152779132633, −3.40975102119229741386823549698, −2.74397258160590768831714217525, −2.31190807247247946579923890637, −1.53690578454560135712324081179, −1.17228886242971314309990552812,
1.17228886242971314309990552812, 1.53690578454560135712324081179, 2.31190807247247946579923890637, 2.74397258160590768831714217525, 3.40975102119229741386823549698, 4.26099745037811799152779132633, 4.64385789642771307621757561964, 4.94700711009357496864534098698, 5.39852558347754735104108081797, 5.97398016747507603353200514102, 6.73632075155858562443002999867, 7.11770500842890259526047007230, 7.34702615440567392596857320571, 8.112627193890949061808269468339, 8.198649989471353669686261061004
