| L(s) = 1 | − 9·9-s + 52·19-s + 92·31-s + 94·49-s + 148·61-s − 284·79-s + 81·81-s + 428·109-s + 242·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 146·169-s − 468·171-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | − 9-s + 2.73·19-s + 2.96·31-s + 1.91·49-s + 2.42·61-s − 3.59·79-s + 81-s + 3.92·109-s + 2·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 0.863·169-s − 2.73·171-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.418903137\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.418903137\) |
| \(L(2)\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 94 T^{2} + p^{4} T^{4} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 146 T^{2} + p^{4} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 26 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 46 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 2062 T^{2} + p^{4} T^{4} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 3214 T^{2} + p^{4} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 74 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 5906 T^{2} + p^{4} T^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 8542 T^{2} + p^{4} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 142 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 18814 T^{2} + p^{4} T^{4} \) |
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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
−9.880537737816700097522612741882, −9.379230643072970718762711422715, −8.811834134187798872657099847090, −8.624231399562789067105406458078, −8.162868764448231638176178227161, −7.73818475963099160195775975994, −7.26987969447683356154256588877, −6.97704308632355725154152325984, −6.40341970253319243018242083721, −5.89328273971928610717249220326, −5.43150070137522414738470990804, −5.35363652451805977746735975063, −4.47983358646155100819597314140, −4.30133988471074141101764734825, −3.35229523741576109504318034474, −3.10180569049129699416252087237, −2.67359654634481488137121858420, −1.97087687536586760557602191897, −0.912761532467478267400477492868, −0.74520079872564095509918015872,
0.74520079872564095509918015872, 0.912761532467478267400477492868, 1.97087687536586760557602191897, 2.67359654634481488137121858420, 3.10180569049129699416252087237, 3.35229523741576109504318034474, 4.30133988471074141101764734825, 4.47983358646155100819597314140, 5.35363652451805977746735975063, 5.43150070137522414738470990804, 5.89328273971928610717249220326, 6.40341970253319243018242083721, 6.97704308632355725154152325984, 7.26987969447683356154256588877, 7.73818475963099160195775975994, 8.162868764448231638176178227161, 8.624231399562789067105406458078, 8.811834134187798872657099847090, 9.379230643072970718762711422715, 9.880537737816700097522612741882
