| L(s) = 1 | − 3-s − 2·7-s + 9-s − 4·11-s + 13-s + 6·17-s − 6·19-s + 2·21-s − 27-s + 2·29-s − 6·31-s + 4·33-s + 10·37-s − 39-s + 8·41-s + 12·43-s − 12·47-s − 3·49-s − 6·51-s − 6·53-s + 6·57-s − 2·61-s − 2·63-s + 2·67-s − 8·71-s − 14·73-s + 8·77-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.755·7-s + 1/3·9-s − 1.20·11-s + 0.277·13-s + 1.45·17-s − 1.37·19-s + 0.436·21-s − 0.192·27-s + 0.371·29-s − 1.07·31-s + 0.696·33-s + 1.64·37-s − 0.160·39-s + 1.24·41-s + 1.82·43-s − 1.75·47-s − 3/7·49-s − 0.840·51-s − 0.824·53-s + 0.794·57-s − 0.256·61-s − 0.251·63-s + 0.244·67-s − 0.949·71-s − 1.63·73-s + 0.911·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−14.59015221796410, −14.09236439613473, −13.19439899862283, −12.89178871240395, −12.72633918062530, −12.09267002740593, −11.40309442392165, −10.87960205431942, −10.55535333591948, −9.966764859200370, −9.527109535061026, −8.978814108035441, −8.169294102078076, −7.751288088072538, −7.338012420266175, −6.500242305724634, −6.007615723791314, −5.748227191561168, −4.961776203375874, −4.421796811251976, −3.753384075285823, −3.030645457454382, −2.539790790261237, −1.650541047243456, −0.7591598109468131, 0,
0.7591598109468131, 1.650541047243456, 2.539790790261237, 3.030645457454382, 3.753384075285823, 4.421796811251976, 4.961776203375874, 5.748227191561168, 6.007615723791314, 6.500242305724634, 7.338012420266175, 7.751288088072538, 8.169294102078076, 8.978814108035441, 9.527109535061026, 9.966764859200370, 10.55535333591948, 10.87960205431942, 11.40309442392165, 12.09267002740593, 12.72633918062530, 12.89178871240395, 13.19439899862283, 14.09236439613473, 14.59015221796410
