| L(s) = 1 | + 4·7-s + 4·25-s + 32·37-s − 8·43-s − 2·49-s + 32·67-s + 16·79-s − 16·109-s + 8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4·169-s + 173-s + 16·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | + 1.51·7-s + 4/5·25-s + 5.26·37-s − 1.21·43-s − 2/7·49-s + 3.90·67-s + 1.80·79-s − 1.53·109-s + 8/11·121-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 + 4/13·169-s + 0.0760·173-s + 1.20·175-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.362616355\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.362616355\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| good | 5 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 28 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 40 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 + 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 56 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 - 124 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 122 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 118 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 170 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−5.86130029350441993633685984980, −5.78822721355972984641798936482, −5.78506192910887284201842792161, −5.16814440814091050452836183426, −4.97025084353375584785565688759, −4.93606654516407299848199225051, −4.88163990796684336707337830835, −4.82545676169597095628216853129, −4.26089802773000417491322999719, −4.19348234646654936414351659543, −4.05537498601006939298680794673, −3.73394319421718863815917825658, −3.58941020542083289872623269653, −3.36603570331381972522467523265, −2.94335505012381750539994897876, −2.78885286239999672440940975005, −2.52986387098497523754355384376, −2.38599325909813602018717448420, −2.15089762282118133501820677163, −1.91168199655078263376519365729, −1.44950482660292942147825198659, −1.30140248096369415620708328802, −0.977914841770974440464099890936, −0.75311999981005301081181906987, −0.38412651392809865137391281520,
0.38412651392809865137391281520, 0.75311999981005301081181906987, 0.977914841770974440464099890936, 1.30140248096369415620708328802, 1.44950482660292942147825198659, 1.91168199655078263376519365729, 2.15089762282118133501820677163, 2.38599325909813602018717448420, 2.52986387098497523754355384376, 2.78885286239999672440940975005, 2.94335505012381750539994897876, 3.36603570331381972522467523265, 3.58941020542083289872623269653, 3.73394319421718863815917825658, 4.05537498601006939298680794673, 4.19348234646654936414351659543, 4.26089802773000417491322999719, 4.82545676169597095628216853129, 4.88163990796684336707337830835, 4.93606654516407299848199225051, 4.97025084353375584785565688759, 5.16814440814091050452836183426, 5.78506192910887284201842792161, 5.78822721355972984641798936482, 5.86130029350441993633685984980
