| L(s) = 1 | − 24·13-s + 4·25-s − 32·37-s − 2·49-s − 8·61-s + 8·73-s − 24·97-s − 16·109-s − 40·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 308·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | − 6.65·13-s + 4/5·25-s − 5.26·37-s − 2/7·49-s − 1.02·61-s + 0.936·73-s − 2.43·97-s − 1.53·109-s − 3.63·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 23.6·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 5 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 44 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 40 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 104 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 89 | $C_2^2$ | \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
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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
−6.56838046709271704416192446899, −6.20099601355347731446877621802, −5.92962879205708946777875567888, −5.45246354739394560912715681948, −5.39883488010832727775566639967, −5.25358561188571956025936718792, −5.15109049371457831039649020513, −5.04015736137516009135718947757, −5.01782713062812631051384436736, −4.56117198366236513425591792397, −4.43129945513003024825216650134, −4.27542148405272055962733025667, −4.14571678259333694894240193715, −3.60493391855988244187056645824, −3.37965203909395327447968569863, −3.34038358411336735405385424237, −3.02457742325209526002657234705, −2.66243303672476388568699677906, −2.55863562662590887780279515853, −2.44931022475125646472789536173, −2.16915247958431223576993142423, −1.86191165156284044426051644246, −1.85662597664469685605532191669, −1.19535508099653498395602079920, −1.10296259542620868119074977817, 0, 0, 0, 0,
1.10296259542620868119074977817, 1.19535508099653498395602079920, 1.85662597664469685605532191669, 1.86191165156284044426051644246, 2.16915247958431223576993142423, 2.44931022475125646472789536173, 2.55863562662590887780279515853, 2.66243303672476388568699677906, 3.02457742325209526002657234705, 3.34038358411336735405385424237, 3.37965203909395327447968569863, 3.60493391855988244187056645824, 4.14571678259333694894240193715, 4.27542148405272055962733025667, 4.43129945513003024825216650134, 4.56117198366236513425591792397, 5.01782713062812631051384436736, 5.04015736137516009135718947757, 5.15109049371457831039649020513, 5.25358561188571956025936718792, 5.39883488010832727775566639967, 5.45246354739394560912715681948, 5.92962879205708946777875567888, 6.20099601355347731446877621802, 6.56838046709271704416192446899
