| L(s) = 1 | + 2·3-s − 4-s + 3·9-s − 2·12-s + 16-s − 12·17-s − 25-s + 4·27-s − 12·29-s − 3·36-s + 8·43-s + 2·48-s − 2·49-s − 24·51-s − 12·53-s − 20·61-s − 64-s + 12·68-s − 2·75-s + 16·79-s + 5·81-s − 24·87-s + 100-s − 36·101-s + 8·103-s − 24·107-s − 4·108-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1/2·4-s + 9-s − 0.577·12-s + 1/4·16-s − 2.91·17-s − 1/5·25-s + 0.769·27-s − 2.22·29-s − 1/2·36-s + 1.21·43-s + 0.288·48-s − 2/7·49-s − 3.36·51-s − 1.64·53-s − 2.56·61-s − 1/8·64-s + 1.45·68-s − 0.230·75-s + 1.80·79-s + 5/9·81-s − 2.57·87-s + 1/10·100-s − 3.58·101-s + 0.788·103-s − 2.32·107-s − 0.384·108-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25704900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25704900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| 13 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 146 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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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
−8.127556002279082142514529988207, −7.73562877633696931316770037680, −7.38060663391904745492757492819, −7.16198049135809868549882999993, −6.58050436458872900406455748756, −6.30364501080472299132614678930, −6.06106737387786767843239252016, −5.39791813652783019594829251342, −4.87711270690896588078666881478, −4.81585939898988208509292805131, −4.08369078714151938323789688344, −3.97252259699027145899570373537, −3.71790943787385589931687958200, −2.96959076936302955068544631647, −2.54550686686188345017479747572, −2.33400265261670445054979910362, −1.60991187752735837712633891342, −1.40311373771204694994124485920, 0, 0,
1.40311373771204694994124485920, 1.60991187752735837712633891342, 2.33400265261670445054979910362, 2.54550686686188345017479747572, 2.96959076936302955068544631647, 3.71790943787385589931687958200, 3.97252259699027145899570373537, 4.08369078714151938323789688344, 4.81585939898988208509292805131, 4.87711270690896588078666881478, 5.39791813652783019594829251342, 6.06106737387786767843239252016, 6.30364501080472299132614678930, 6.58050436458872900406455748756, 7.16198049135809868549882999993, 7.38060663391904745492757492819, 7.73562877633696931316770037680, 8.127556002279082142514529988207
