| L(s) = 1 | + 2-s + 4-s − 5-s + 2·8-s − 10-s + 2·16-s − 2·17-s − 2·19-s − 20-s + 23-s + 31-s + 2·32-s − 2·34-s − 2·38-s − 2·40-s + 46-s − 2·47-s − 49-s − 2·53-s + 61-s + 62-s + 3·64-s − 2·68-s − 2·76-s + 79-s − 2·80-s + 83-s + ⋯ |
| L(s) = 1 | + 2-s + 4-s − 5-s + 2·8-s − 10-s + 2·16-s − 2·17-s − 2·19-s − 20-s + 23-s + 31-s + 2·32-s − 2·34-s − 2·38-s − 2·40-s + 46-s − 2·47-s − 49-s − 2·53-s + 61-s + 62-s + 3·64-s − 2·68-s − 2·76-s + 79-s − 2·80-s + 83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.114883543\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.114883543\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 2 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 83 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
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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
−11.46669758665371225287517015186, −11.37914445742390888364191252848, −10.82659253995829168080045524512, −10.78043409833624124403047556995, −10.02975199629696553070240282299, −9.569657504300470819616103727247, −8.609836530904373440437456688220, −8.567094889802317980942082513128, −7.73355615850760394401301738471, −7.70797103589107901473095663846, −6.70128422669499857385229090308, −6.57761387172952325273896913430, −6.28696348819327222079394110170, −5.01996500504280180830195282942, −4.88534069693732128461046620751, −4.30746236260083919008629593266, −3.96283284314292897956239151632, −3.20070404887720902396814416450, −2.36470940730744213209832599340, −1.72241182277389574123742867895,
1.72241182277389574123742867895, 2.36470940730744213209832599340, 3.20070404887720902396814416450, 3.96283284314292897956239151632, 4.30746236260083919008629593266, 4.88534069693732128461046620751, 5.01996500504280180830195282942, 6.28696348819327222079394110170, 6.57761387172952325273896913430, 6.70128422669499857385229090308, 7.70797103589107901473095663846, 7.73355615850760394401301738471, 8.567094889802317980942082513128, 8.609836530904373440437456688220, 9.569657504300470819616103727247, 10.02975199629696553070240282299, 10.78043409833624124403047556995, 10.82659253995829168080045524512, 11.37914445742390888364191252848, 11.46669758665371225287517015186
