L(s) = 1 | − 3-s + 3·5-s − 3·7-s + 9-s + 11-s + 2·13-s − 3·15-s − 5·17-s − 19-s + 3·21-s − 4·23-s + 4·25-s − 27-s + 6·29-s − 2·31-s − 33-s − 9·35-s − 8·37-s − 2·39-s − 8·41-s − 13·43-s + 3·45-s + 13·47-s + 2·49-s + 5·51-s + 6·53-s + 3·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1.34·5-s − 1.13·7-s + 1/3·9-s + 0.301·11-s + 0.554·13-s − 0.774·15-s − 1.21·17-s − 0.229·19-s + 0.654·21-s − 0.834·23-s + 4/5·25-s − 0.192·27-s + 1.11·29-s − 0.359·31-s − 0.174·33-s − 1.52·35-s − 1.31·37-s − 0.320·39-s − 1.24·41-s − 1.98·43-s + 0.447·45-s + 1.89·47-s + 2/7·49-s + 0.700·51-s + 0.824·53-s + 0.404·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 19 | \( 1 + T \) |
good | 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 13 T + p T^{2} \) |
| 47 | \( 1 - 13 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−8.453235268768676142268982537470, −6.91416562298774471515672731053, −6.69012650769339551249472044522, −5.97055699321514567874636861720, −5.39633328895611499844939407246, −4.37481346272208907658412133211, −3.45316860614778839296918966250, −2.38163485579970949069695588350, −1.50121656193568560718071555833, 0,
1.50121656193568560718071555833, 2.38163485579970949069695588350, 3.45316860614778839296918966250, 4.37481346272208907658412133211, 5.39633328895611499844939407246, 5.97055699321514567874636861720, 6.69012650769339551249472044522, 6.91416562298774471515672731053, 8.453235268768676142268982537470