| L(s) = 1 | + 2·3-s + 4·5-s + 2·9-s + 6·11-s − 6·13-s + 8·15-s + 2·19-s − 16·23-s + 11·25-s + 6·27-s + 6·29-s + 12·33-s − 6·37-s − 12·39-s + 6·43-s + 8·45-s − 14·49-s + 18·53-s + 24·55-s + 4·57-s − 18·59-s − 10·61-s − 24·65-s − 6·67-s − 32·69-s − 12·73-s + 22·75-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1.78·5-s + 2/3·9-s + 1.80·11-s − 1.66·13-s + 2.06·15-s + 0.458·19-s − 3.33·23-s + 11/5·25-s + 1.15·27-s + 1.11·29-s + 2.08·33-s − 0.986·37-s − 1.92·39-s + 0.914·43-s + 1.19·45-s − 2·49-s + 2.47·53-s + 3.23·55-s + 0.529·57-s − 2.34·59-s − 1.28·61-s − 2.97·65-s − 0.733·67-s − 3.85·69-s − 1.40·73-s + 2.54·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.162629925\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.162629925\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 90 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 18 T + 162 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 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
−11.85284655266805570811069939811, −11.76200344151940049469599825859, −10.48922300035016758838743428643, −10.33169219286481316322192022941, −9.959551598644471291708644176229, −9.433203755690194327866007262722, −9.176225447303178520403403656083, −8.818028608068215222624588998115, −8.060284106985485844780167668018, −7.75264263667069021427356344867, −6.81560548147572003152009015023, −6.71311471301466237711250285079, −5.90978916390900229679808612474, −5.67303222699339124161223373015, −4.52408837620271187859619395269, −4.43851807196688628900913412284, −3.36299286050689698365127824960, −2.74693805849769539429696274380, −2.02726751140973095117906886361, −1.55406267284351505026766380256,
1.55406267284351505026766380256, 2.02726751140973095117906886361, 2.74693805849769539429696274380, 3.36299286050689698365127824960, 4.43851807196688628900913412284, 4.52408837620271187859619395269, 5.67303222699339124161223373015, 5.90978916390900229679808612474, 6.71311471301466237711250285079, 6.81560548147572003152009015023, 7.75264263667069021427356344867, 8.060284106985485844780167668018, 8.818028608068215222624588998115, 9.176225447303178520403403656083, 9.433203755690194327866007262722, 9.959551598644471291708644176229, 10.33169219286481316322192022941, 10.48922300035016758838743428643, 11.76200344151940049469599825859, 11.85284655266805570811069939811
