L(s) = 1 | + 2-s + 4-s + 4·5-s + 7-s + 8-s + 9-s + 4·10-s − 8·11-s − 12·13-s + 14-s + 16-s + 18-s + 4·20-s − 8·22-s + 2·25-s − 12·26-s + 28-s + 32-s + 4·35-s + 36-s + 4·40-s − 8·43-s − 8·44-s + 4·45-s + 49-s + 2·50-s − 12·52-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.78·5-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 1.26·10-s − 2.41·11-s − 3.32·13-s + 0.267·14-s + 1/4·16-s + 0.235·18-s + 0.894·20-s − 1.70·22-s + 2/5·25-s − 2.35·26-s + 0.188·28-s + 0.176·32-s + 0.676·35-s + 1/6·36-s + 0.632·40-s − 1.21·43-s − 1.20·44-s + 0.596·45-s + 1/7·49-s + 0.282·50-s − 1.66·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 395136 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 395136 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_1$ | \( 1 - T \) |
good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−8.186632228453794414425905803942, −7.68532191797102726979476428936, −7.68501515018855362022465153557, −6.95470194591461271128583103192, −6.60807660052202573549483918899, −5.76536999919480153066315644301, −5.47990216984766844713395370376, −5.11740288034969242991482089170, −4.82857955948335988380640999352, −4.30167637849672643861137972376, −3.08481257245522303063597583848, −2.66792336770788585023270622114, −2.16063348217202322480816158281, −1.88351369279783613545674963411, 0,
1.88351369279783613545674963411, 2.16063348217202322480816158281, 2.66792336770788585023270622114, 3.08481257245522303063597583848, 4.30167637849672643861137972376, 4.82857955948335988380640999352, 5.11740288034969242991482089170, 5.47990216984766844713395370376, 5.76536999919480153066315644301, 6.60807660052202573549483918899, 6.95470194591461271128583103192, 7.68501515018855362022465153557, 7.68532191797102726979476428936, 8.186632228453794414425905803942