L(s) = 1 | − 2-s + 2·3-s + 4-s − 2·6-s − 8-s − 3·9-s + 2·12-s + 16-s + 3·18-s + 4·19-s − 2·24-s − 25-s − 14·27-s − 32-s − 3·36-s − 4·38-s − 24·41-s + 16·43-s + 2·48-s − 13·49-s + 50-s + 14·54-s + 8·57-s − 18·59-s + 64-s − 8·67-s + 3·72-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.816·6-s − 0.353·8-s − 9-s + 0.577·12-s + 1/4·16-s + 0.707·18-s + 0.917·19-s − 0.408·24-s − 1/5·25-s − 2.69·27-s − 0.176·32-s − 1/2·36-s − 0.648·38-s − 3.74·41-s + 2.43·43-s + 0.288·48-s − 1.85·49-s + 0.141·50-s + 1.90·54-s + 1.05·57-s − 2.34·59-s + 1/8·64-s − 0.977·67-s + 0.353·72-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 798848 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798848 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.452304801\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.452304801\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 17 T + p T^{2} )^{2} \) |
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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
−8.083630712180759633706240971225, −8.008923035003699396981349994067, −7.63212533820846826900119928753, −7.13501126218469989101311707707, −6.43698619599801084891178808154, −6.11894230966477053103157726414, −5.66674862285945396423871921055, −4.96602444926117376624174078421, −4.68407228651939958115867257852, −3.50543676040170862896935659510, −3.35403695021279589879198759951, −3.05075526863841122894839780512, −2.07130866665970371265555871793, −1.90147284566448759002651582081, −0.57937075713009226440087094657,
0.57937075713009226440087094657, 1.90147284566448759002651582081, 2.07130866665970371265555871793, 3.05075526863841122894839780512, 3.35403695021279589879198759951, 3.50543676040170862896935659510, 4.68407228651939958115867257852, 4.96602444926117376624174078421, 5.66674862285945396423871921055, 6.11894230966477053103157726414, 6.43698619599801084891178808154, 7.13501126218469989101311707707, 7.63212533820846826900119928753, 8.008923035003699396981349994067, 8.083630712180759633706240971225