| L(s) = 1 | − 3-s − 5-s − 2·9-s − 2·11-s + 15-s − 10·23-s − 4·25-s + 2·27-s + 10·31-s + 2·33-s − 10·37-s + 2·45-s − 16·47-s + 8·49-s − 4·53-s + 2·55-s − 4·59-s + 2·67-s + 10·69-s − 16·71-s + 4·75-s + 7·81-s + 22·89-s − 10·93-s − 14·97-s + 4·99-s − 2·103-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 2/3·9-s − 0.603·11-s + 0.258·15-s − 2.08·23-s − 4/5·25-s + 0.384·27-s + 1.79·31-s + 0.348·33-s − 1.64·37-s + 0.298·45-s − 2.33·47-s + 8/7·49-s − 0.549·53-s + 0.269·55-s − 0.520·59-s + 0.244·67-s + 1.20·69-s − 1.89·71-s + 0.461·75-s + 7/9·81-s + 2.33·89-s − 1.03·93-s − 1.42·97-s + 0.402·99-s − 0.197·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29040 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29040 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
| 11 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 24 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
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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
−10.36374883997169791164214822460, −9.906566539530146269782375770738, −9.297414656799504870432131549516, −8.488460286034259992824918190629, −8.130599524009900109543910465446, −7.75312156167697894020424745322, −6.95986755461883431688224976728, −6.22176796833594975251026998923, −5.91893991817109371697024201968, −5.17178128346823184568413524452, −4.56145089833578904943988733388, −3.77052797663273840924601578362, −2.98711440222433410761523446210, −1.94297767047486301309072781769, 0,
1.94297767047486301309072781769, 2.98711440222433410761523446210, 3.77052797663273840924601578362, 4.56145089833578904943988733388, 5.17178128346823184568413524452, 5.91893991817109371697024201968, 6.22176796833594975251026998923, 6.95986755461883431688224976728, 7.75312156167697894020424745322, 8.130599524009900109543910465446, 8.488460286034259992824918190629, 9.297414656799504870432131549516, 9.906566539530146269782375770738, 10.36374883997169791164214822460
