| L(s) = 1 | − 2·2-s + 3·4-s − 4·8-s − 9-s + 5·16-s − 2·17-s + 2·18-s − 25-s − 6·32-s + 4·34-s − 3·36-s + 4·47-s − 2·49-s + 2·50-s + 7·64-s − 6·68-s + 4·72-s + 81-s − 8·94-s + 4·98-s − 3·100-s − 2·121-s + 127-s − 8·128-s + 131-s + 8·136-s + 137-s + ⋯ |
| L(s) = 1 | − 2·2-s + 3·4-s − 4·8-s − 9-s + 5·16-s − 2·17-s + 2·18-s − 25-s − 6·32-s + 4·34-s − 3·36-s + 4·47-s − 2·49-s + 2·50-s + 7·64-s − 6·68-s + 4·72-s + 81-s − 8·94-s + 4·98-s − 3·100-s − 2·121-s + 127-s − 8·128-s + 131-s + 8·136-s + 137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4161600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4161600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2841021297\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2841021297\) |
| \(L(1)\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| 17 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 47 | $C_1$ | \( ( 1 - T )^{4} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
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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
−9.862247532349353100599198105352, −9.198237556737216374530623321019, −8.937338730127682590102136482179, −8.319181700244944463799992017180, −8.208062461854400870956281798795, −7.56848596346261069070412965320, −7.47179006703923904991620179878, −6.72466227307913697621380941399, −6.69613205757517073993013245611, −6.02859335694474862754770264475, −5.90002311286312899972774297185, −5.34681013654147847007234239427, −4.76595554897815050849615463926, −3.95601988620512104240189399943, −3.69984987874954663982115755958, −2.73059760688896385685619785822, −2.65880406357845219204083894496, −2.08734021763495813816948097568, −1.52505240288990944838480689011, −0.51293976111304546416455528160,
0.51293976111304546416455528160, 1.52505240288990944838480689011, 2.08734021763495813816948097568, 2.65880406357845219204083894496, 2.73059760688896385685619785822, 3.69984987874954663982115755958, 3.95601988620512104240189399943, 4.76595554897815050849615463926, 5.34681013654147847007234239427, 5.90002311286312899972774297185, 6.02859335694474862754770264475, 6.69613205757517073993013245611, 6.72466227307913697621380941399, 7.47179006703923904991620179878, 7.56848596346261069070412965320, 8.208062461854400870956281798795, 8.319181700244944463799992017180, 8.937338730127682590102136482179, 9.198237556737216374530623321019, 9.862247532349353100599198105352
