L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s − 12·11-s − 12-s − 4·13-s + 16-s − 18-s + 12·22-s − 2·23-s + 24-s − 6·25-s + 4·26-s − 27-s − 32-s + 12·33-s + 36-s + 4·39-s − 12·44-s + 2·46-s − 16·47-s − 48-s − 10·49-s + 6·50-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 3.61·11-s − 0.288·12-s − 1.10·13-s + 1/4·16-s − 0.235·18-s + 2.55·22-s − 0.417·23-s + 0.204·24-s − 6/5·25-s + 0.784·26-s − 0.192·27-s − 0.176·32-s + 2.08·33-s + 1/6·36-s + 0.640·39-s − 1.80·44-s + 0.294·46-s − 2.33·47-s − 0.144·48-s − 1.42·49-s + 0.848·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 457056 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 457056 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 2 | $C_1$ | \( 1 + T \) |
| 3 | $C_1$ | \( 1 + T \) |
| 23 | $C_1$ | \( ( 1 + T )^{2} \) |
good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
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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
−8.009088544194373664704576673472, −7.74940445665935159391608281760, −7.48954786032446015166944686975, −6.81404055785242482314012546704, −6.25250569926844252670184221360, −5.70446099983308366623422114205, −5.34329120767643148260995938996, −4.78570582427944795062347753374, −4.61781233004508478544485198077, −3.38807414963777119141033594454, −2.91631229938310047782795220333, −2.33137889789155181282145246786, −1.75673035985210661335841948053, 0, 0,
1.75673035985210661335841948053, 2.33137889789155181282145246786, 2.91631229938310047782795220333, 3.38807414963777119141033594454, 4.61781233004508478544485198077, 4.78570582427944795062347753374, 5.34329120767643148260995938996, 5.70446099983308366623422114205, 6.25250569926844252670184221360, 6.81404055785242482314012546704, 7.48954786032446015166944686975, 7.74940445665935159391608281760, 8.009088544194373664704576673472