| L(s) = 1 | + 2·5-s − 6·9-s + 12·13-s − 12·17-s + 3·25-s − 4·29-s − 20·37-s + 20·41-s − 12·45-s + 2·49-s − 20·53-s − 4·61-s + 24·65-s − 28·73-s + 27·81-s − 24·85-s − 12·89-s + 4·97-s − 20·101-s − 4·109-s + 4·113-s − 72·117-s + 121-s + 4·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 2·9-s + 3.32·13-s − 2.91·17-s + 3/5·25-s − 0.742·29-s − 3.28·37-s + 3.12·41-s − 1.78·45-s + 2/7·49-s − 2.74·53-s − 0.512·61-s + 2.97·65-s − 3.27·73-s + 3·81-s − 2.60·85-s − 1.27·89-s + 0.406·97-s − 1.99·101-s − 0.383·109-s + 0.376·113-s − 6.65·117-s + 1/11·121-s + 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 387200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−8.613808585244655340820053101754, −8.294068680294176403765781809942, −7.58115128738550044456594424531, −6.79637909196263478888408793326, −6.28118370578645402196158692333, −6.27108685587236300377566499608, −5.63628573418698799390682191313, −5.40941065075534095194244033749, −4.42524465686514871694159803405, −4.05902436868145020817401928937, −3.26973932766649133809304477558, −2.88180193076351113983639438164, −2.03028148171803176729508478145, −1.46303061533755639665183373410, 0,
1.46303061533755639665183373410, 2.03028148171803176729508478145, 2.88180193076351113983639438164, 3.26973932766649133809304477558, 4.05902436868145020817401928937, 4.42524465686514871694159803405, 5.40941065075534095194244033749, 5.63628573418698799390682191313, 6.27108685587236300377566499608, 6.28118370578645402196158692333, 6.79637909196263478888408793326, 7.58115128738550044456594424531, 8.294068680294176403765781809942, 8.613808585244655340820053101754
