| L(s) = 1 | − 3-s − 7-s − 2·13-s + 19-s + 21-s − 25-s + 27-s + 31-s + 37-s + 2·39-s − 2·43-s − 57-s − 2·61-s + 67-s + 73-s + 75-s + 79-s − 81-s + 2·91-s − 93-s + 4·97-s + 103-s + 109-s − 111-s − 121-s + 127-s + 2·129-s + ⋯ |
| L(s) = 1 | − 3-s − 7-s − 2·13-s + 19-s + 21-s − 25-s + 27-s + 31-s + 37-s + 2·39-s − 2·43-s − 57-s − 2·61-s + 67-s + 73-s + 75-s + 79-s − 81-s + 2·91-s − 93-s + 4·97-s + 103-s + 109-s − 111-s − 121-s + 127-s + 2·129-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2070897827\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2070897827\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 97 | $C_1$ | \( ( 1 - T )^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.74600246740631530719188745074, −14.33197631703146060715751911730, −13.63570593099300485737171669080, −13.20498120399190008275121892141, −12.33651690769049716182029767608, −12.27926998190461195602564601842, −11.57081641310592920438165715557, −11.28616606617349332842095808165, −10.11348932277312344731628512468, −10.10977840826758241325303230439, −9.514310179830675798001626081592, −8.796010299104171695587299127572, −7.74268924650426005669373898019, −7.44725460239841168931737259170, −6.33268371238830377081639818438, −6.30895696473354994681334896900, −5.07920460323713432335456360487, −4.89098280278214127535463544377, −3.52084052442906496218604031148, −2.56497434085969418732679777437,
2.56497434085969418732679777437, 3.52084052442906496218604031148, 4.89098280278214127535463544377, 5.07920460323713432335456360487, 6.30895696473354994681334896900, 6.33268371238830377081639818438, 7.44725460239841168931737259170, 7.74268924650426005669373898019, 8.796010299104171695587299127572, 9.514310179830675798001626081592, 10.10977840826758241325303230439, 10.11348932277312344731628512468, 11.28616606617349332842095808165, 11.57081641310592920438165715557, 12.27926998190461195602564601842, 12.33651690769049716182029767608, 13.20498120399190008275121892141, 13.63570593099300485737171669080, 14.33197631703146060715751911730, 14.74600246740631530719188745074
