| L(s) = 1 | + 2·3-s − 3·9-s + 6·11-s + 6·17-s − 2·19-s − 14·27-s + 12·33-s + 18·41-s + 8·43-s − 2·49-s + 12·51-s − 4·57-s − 24·59-s + 22·67-s − 14·73-s − 4·81-s + 30·83-s + 6·89-s + 28·97-s − 18·99-s + 18·107-s − 30·113-s + 5·121-s + 36·123-s + 127-s + 16·129-s + 131-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 9-s + 1.80·11-s + 1.45·17-s − 0.458·19-s − 2.69·27-s + 2.08·33-s + 2.81·41-s + 1.21·43-s − 2/7·49-s + 1.68·51-s − 0.529·57-s − 3.12·59-s + 2.68·67-s − 1.63·73-s − 4/9·81-s + 3.29·83-s + 0.635·89-s + 2.84·97-s − 1.80·99-s + 1.74·107-s − 2.82·113-s + 5/11·121-s + 3.24·123-s + 0.0887·127-s + 1.40·129-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40960000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.201384900\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.201384900\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
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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
−8.092215483860948624802294134786, −7.77103803940088625522562349480, −7.60334690251796714033122268217, −7.55075724730991410678530337332, −6.58313598318234900386278080485, −6.48762318905448830540571611019, −6.03131544470245267037028427810, −5.93646480763440410149302484187, −5.35539635736104099358611396187, −5.03995983879745872141396852651, −4.42917273509330128606505983426, −4.07577574982298080321030268756, −3.61789437003575579512958960590, −3.53982147367611561777893790859, −2.82141262863596428801118183360, −2.78210057972701347165061188507, −2.08181765179382447060208515786, −1.75753263798141382406855779200, −1.01051541151638899632887973399, −0.59761585589520308526573817323,
0.59761585589520308526573817323, 1.01051541151638899632887973399, 1.75753263798141382406855779200, 2.08181765179382447060208515786, 2.78210057972701347165061188507, 2.82141262863596428801118183360, 3.53982147367611561777893790859, 3.61789437003575579512958960590, 4.07577574982298080321030268756, 4.42917273509330128606505983426, 5.03995983879745872141396852651, 5.35539635736104099358611396187, 5.93646480763440410149302484187, 6.03131544470245267037028427810, 6.48762318905448830540571611019, 6.58313598318234900386278080485, 7.55075724730991410678530337332, 7.60334690251796714033122268217, 7.77103803940088625522562349480, 8.092215483860948624802294134786
