L(s) = 1 | + 2·2-s + 3-s − 4-s + 2·6-s − 8·8-s + 9-s − 12-s − 7·16-s + 2·18-s − 19-s − 8·24-s − 6·25-s + 27-s + 4·29-s + 14·32-s − 36-s − 2·38-s − 4·41-s − 8·43-s − 7·48-s − 14·49-s − 12·50-s − 12·53-s + 2·54-s − 57-s + 8·58-s − 24·59-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 0.577·3-s − 1/2·4-s + 0.816·6-s − 2.82·8-s + 1/3·9-s − 0.288·12-s − 7/4·16-s + 0.471·18-s − 0.229·19-s − 1.63·24-s − 6/5·25-s + 0.192·27-s + 0.742·29-s + 2.47·32-s − 1/6·36-s − 0.324·38-s − 0.624·41-s − 1.21·43-s − 1.01·48-s − 2·49-s − 1.69·50-s − 1.64·53-s + 0.272·54-s − 0.132·57-s + 1.05·58-s − 3.12·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 185193 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185193 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 3 | $C_1$ | \( 1 - T \) |
| 19 | $C_1$ | \( 1 + T \) |
good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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
−9.032116632714081125535639739797, −8.263513432862467471567669978548, −8.149677875507257955940242249903, −7.61675340778570823466262486344, −6.50584246724891866846885766490, −6.46787589567348650226556085189, −5.82020929693792166281675138030, −5.18025942388109829342113033634, −4.59000163102556089132067894166, −4.56991321080597356703354311172, −3.52439250757221266883814274902, −3.44365518362789223021999993571, −2.73578806708278678142126219562, −1.66141629985644983818808667860, 0,
1.66141629985644983818808667860, 2.73578806708278678142126219562, 3.44365518362789223021999993571, 3.52439250757221266883814274902, 4.56991321080597356703354311172, 4.59000163102556089132067894166, 5.18025942388109829342113033634, 5.82020929693792166281675138030, 6.46787589567348650226556085189, 6.50584246724891866846885766490, 7.61675340778570823466262486344, 8.149677875507257955940242249903, 8.263513432862467471567669978548, 9.032116632714081125535639739797