| L(s) = 1 | − 2·4-s + 8·7-s + 3·16-s + 4·25-s − 16·28-s − 8·37-s − 44·43-s + 34·49-s − 4·64-s − 28·67-s + 32·79-s − 8·100-s + 16·109-s + 24·112-s − 28·121-s + 127-s + 131-s + 137-s + 139-s + 16·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 46·169-s + 88·172-s + ⋯ |
| L(s) = 1 | − 4-s + 3.02·7-s + 3/4·16-s + 4/5·25-s − 3.02·28-s − 1.31·37-s − 6.70·43-s + 34/7·49-s − 1/2·64-s − 3.42·67-s + 3.60·79-s − 4/5·100-s + 1.53·109-s + 2.26·112-s − 2.54·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.31·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3.53·169-s + 6.70·172-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.067271434\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.067271434\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| good | 5 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 31 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 37 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 49 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 35 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 + 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 97 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 43 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 - 133 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 98 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 154 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 151 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 146 T^{2} + p^{2} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.268345140362973909937409676722, −7.982325305409017081479046817808, −7.976214258126713115784085842596, −7.58097140254410139026035015797, −7.39251321746416321140727485722, −7.03983853065696803929709335219, −6.70479405118618531543716796525, −6.50200967289606503170204745064, −6.33805318076580756535377255225, −5.80658807229212780804422961734, −5.32647056895037338950990251867, −5.25192054419838463801047032516, −5.09326916409939077370862112546, −5.00730869776337642163543849441, −4.44823347395466669575215431044, −4.41590517591630275298128708970, −4.25814546414026737499904151892, −3.37595957445088214530734210099, −3.35477266239183925133216579936, −3.26702218997399691608563766513, −2.45996292262262875043062996812, −1.76967742243704503004371937678, −1.65773064102172855830350171016, −1.61915126982252147919498589474, −0.59204550983037465145953103618,
0.59204550983037465145953103618, 1.61915126982252147919498589474, 1.65773064102172855830350171016, 1.76967742243704503004371937678, 2.45996292262262875043062996812, 3.26702218997399691608563766513, 3.35477266239183925133216579936, 3.37595957445088214530734210099, 4.25814546414026737499904151892, 4.41590517591630275298128708970, 4.44823347395466669575215431044, 5.00730869776337642163543849441, 5.09326916409939077370862112546, 5.25192054419838463801047032516, 5.32647056895037338950990251867, 5.80658807229212780804422961734, 6.33805318076580756535377255225, 6.50200967289606503170204745064, 6.70479405118618531543716796525, 7.03983853065696803929709335219, 7.39251321746416321140727485722, 7.58097140254410139026035015797, 7.976214258126713115784085842596, 7.982325305409017081479046817808, 8.268345140362973909937409676722
