| L(s) = 1 | − 2-s − 3·3-s − 2·4-s + 3·6-s − 7-s + 3·8-s + 2·9-s + 2·11-s + 6·12-s − 8·13-s + 14-s + 16-s − 17-s − 2·18-s + 3·21-s − 2·22-s − 3·23-s − 9·24-s + 8·26-s + 6·27-s + 2·28-s − 5·29-s − 6·31-s − 2·32-s − 6·33-s + 34-s − 4·36-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.73·3-s − 4-s + 1.22·6-s − 0.377·7-s + 1.06·8-s + 2/3·9-s + 0.603·11-s + 1.73·12-s − 2.21·13-s + 0.267·14-s + 1/4·16-s − 0.242·17-s − 0.471·18-s + 0.654·21-s − 0.426·22-s − 0.625·23-s − 1.83·24-s + 1.56·26-s + 1.15·27-s + 0.377·28-s − 0.928·29-s − 1.07·31-s − 0.353·32-s − 1.04·33-s + 0.171·34-s − 2/3·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 3 | $D_{4}$ | \( 1 + p T + 7 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 8 T + 37 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + T + 33 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 3 T + 17 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 5 T + 63 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 16 T + 133 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 3 T + 47 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 10 T + 123 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 11 T + 121 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 71 | $C_4$ | \( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 23 T + 277 T^{2} + 23 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 5 T + 153 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 27 T + 337 T^{2} - 27 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 25 T + 303 T^{2} + 25 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + T + 193 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 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
−11.80555197965088696893994809316, −11.11911380295746764466938346288, −10.60287326656920564997447628930, −10.31706666720708864964347944408, −9.666794160679867034356919584764, −9.463444875934616351750339132614, −8.802682250912539586751346681205, −8.576238894185616609405150893583, −7.51500741095694152659535704294, −7.38623110876223889945537992955, −6.66183538805655167913975836344, −6.09098378861924905440480640353, −5.42670884232244898599868521037, −5.16023539937062925692592448713, −4.58869174694174518488131297210, −3.95430665306330740889269912878, −2.96886712006022722229163981006, −1.74053329646528698874613541791, 0, 0,
1.74053329646528698874613541791, 2.96886712006022722229163981006, 3.95430665306330740889269912878, 4.58869174694174518488131297210, 5.16023539937062925692592448713, 5.42670884232244898599868521037, 6.09098378861924905440480640353, 6.66183538805655167913975836344, 7.38623110876223889945537992955, 7.51500741095694152659535704294, 8.576238894185616609405150893583, 8.802682250912539586751346681205, 9.463444875934616351750339132614, 9.666794160679867034356919584764, 10.31706666720708864964347944408, 10.60287326656920564997447628930, 11.11911380295746764466938346288, 11.80555197965088696893994809316
