L(s) = 1 | − 3-s − 5-s + 7-s + 9-s − 3·11-s − 6·13-s + 15-s − 17-s + 6·19-s − 21-s − 4·23-s + 25-s − 27-s + 5·29-s + 9·31-s + 3·33-s − 35-s + 6·39-s − 3·41-s − 7·43-s − 45-s − 6·49-s + 51-s − 53-s + 3·55-s − 6·57-s + 2·59-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.904·11-s − 1.66·13-s + 0.258·15-s − 0.242·17-s + 1.37·19-s − 0.218·21-s − 0.834·23-s + 1/5·25-s − 0.192·27-s + 0.928·29-s + 1.61·31-s + 0.522·33-s − 0.169·35-s + 0.960·39-s − 0.468·41-s − 1.06·43-s − 0.149·45-s − 6/7·49-s + 0.140·51-s − 0.137·53-s + 0.404·55-s − 0.794·57-s + 0.260·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 328560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 328560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 37 | \( 1 \) |
good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 - 3 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−12.76223312696126, −12.08034488470050, −11.96581238323101, −11.65210956944216, −11.12960051903944, −10.38089088147299, −10.15365378839866, −9.873744405072640, −9.282779756821907, −8.564012522882447, −8.150810564340577, −7.669720361973648, −7.391351842658931, −6.840508695730401, −6.297532822449261, −5.778240280108612, −5.064540979605030, −4.841300873448054, −4.593818364328883, −3.764263840897157, −3.088234481916578, −2.685608261062184, −2.072865634964602, −1.350119348010264, −0.6054474036321240, 0,
0.6054474036321240, 1.350119348010264, 2.072865634964602, 2.685608261062184, 3.088234481916578, 3.764263840897157, 4.593818364328883, 4.841300873448054, 5.064540979605030, 5.778240280108612, 6.297532822449261, 6.840508695730401, 7.391351842658931, 7.669720361973648, 8.150810564340577, 8.564012522882447, 9.282779756821907, 9.873744405072640, 10.15365378839866, 10.38089088147299, 11.12960051903944, 11.65210956944216, 11.96581238323101, 12.08034488470050, 12.76223312696126