L(s) = 1 | + 5-s − 11-s + 3·13-s − 3·17-s − 6·19-s − 4·23-s + 25-s − 29-s + 6·31-s + 6·41-s − 6·43-s + 9·47-s − 10·53-s − 55-s − 6·59-s + 3·65-s − 14·67-s − 8·71-s − 6·73-s + 79-s + 12·83-s − 3·85-s + 12·89-s − 6·95-s + 15·97-s + 101-s + 103-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 0.301·11-s + 0.832·13-s − 0.727·17-s − 1.37·19-s − 0.834·23-s + 1/5·25-s − 0.185·29-s + 1.07·31-s + 0.937·41-s − 0.914·43-s + 1.31·47-s − 1.37·53-s − 0.134·55-s − 0.781·59-s + 0.372·65-s − 1.71·67-s − 0.949·71-s − 0.702·73-s + 0.112·79-s + 1.31·83-s − 0.325·85-s + 1.27·89-s − 0.615·95-s + 1.52·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 141120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 141120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 15 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.47423528213662, −13.27278241328788, −12.80757946194115, −12.18764806957152, −11.77936621911972, −11.17561973125263, −10.62198230255297, −10.43786297813191, −9.877458636903562, −9.077435063913754, −8.933128834869305, −8.347356858673332, −7.784475319479826, −7.363832491235544, −6.441044239111429, −6.283681450826496, −5.906174250620793, −5.109193263289464, −4.455065757776426, −4.228023750543397, −3.391968772478779, −2.829802315160523, −2.099688734226595, −1.738224113254436, −0.8209040423253546, 0,
0.8209040423253546, 1.738224113254436, 2.099688734226595, 2.829802315160523, 3.391968772478779, 4.228023750543397, 4.455065757776426, 5.109193263289464, 5.906174250620793, 6.283681450826496, 6.441044239111429, 7.363832491235544, 7.784475319479826, 8.347356858673332, 8.933128834869305, 9.077435063913754, 9.877458636903562, 10.43786297813191, 10.62198230255297, 11.17561973125263, 11.77936621911972, 12.18764806957152, 12.80757946194115, 13.27278241328788, 13.47423528213662