| L(s) = 1 | + 11-s + 2·13-s + 4·17-s − 2·19-s − 5·25-s + 4·29-s + 2·31-s − 37-s − 10·41-s + 2·43-s − 8·47-s − 7·49-s − 10·53-s − 10·61-s + 8·67-s + 16·71-s + 10·73-s + 14·79-s + 4·83-s + 4·89-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.301·11-s + 0.554·13-s + 0.970·17-s − 0.458·19-s − 25-s + 0.742·29-s + 0.359·31-s − 0.164·37-s − 1.56·41-s + 0.304·43-s − 1.16·47-s − 49-s − 1.37·53-s − 1.28·61-s + 0.977·67-s + 1.89·71-s + 1.17·73-s + 1.57·79-s + 0.439·83-s + 0.423·89-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.155653135\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.155653135\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 37 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 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
−14.14553047437173, −13.97685686424062, −13.35986730748899, −12.83694963318444, −12.13420511422619, −12.01230712042395, −11.20601468829360, −10.85435325557063, −10.22126210976186, −9.617883680503312, −9.394171005745981, −8.501024772826130, −8.046384583594267, −7.814953206334100, −6.788895030124097, −6.476867576011621, −5.990553479604434, −5.109205233788917, −4.868727047788554, −3.889608533753564, −3.529866339456466, −2.875689996018159, −1.952819074141322, −1.421802295098129, −0.5115347004896082,
0.5115347004896082, 1.421802295098129, 1.952819074141322, 2.875689996018159, 3.529866339456466, 3.889608533753564, 4.868727047788554, 5.109205233788917, 5.990553479604434, 6.476867576011621, 6.788895030124097, 7.814953206334100, 8.046384583594267, 8.501024772826130, 9.394171005745981, 9.617883680503312, 10.22126210976186, 10.85435325557063, 11.20601468829360, 12.01230712042395, 12.13420511422619, 12.83694963318444, 13.35986730748899, 13.97685686424062, 14.14553047437173
