| L(s) = 1 | + 9·3-s + 20·5-s + 2·7-s + 54·9-s + 52·11-s − 43·13-s + 180·15-s − 50·17-s + 74·19-s + 18·21-s − 23·23-s + 275·25-s + 243·27-s + 7·29-s − 273·31-s + 468·33-s + 40·35-s + 4·37-s − 387·39-s + 123·41-s + 152·43-s + 1.08e3·45-s + 75·47-s − 339·49-s − 450·51-s − 86·53-s + 1.04e3·55-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 1.78·5-s + 0.107·7-s + 2·9-s + 1.42·11-s − 0.917·13-s + 3.09·15-s − 0.713·17-s + 0.893·19-s + 0.187·21-s − 0.208·23-s + 11/5·25-s + 1.73·27-s + 0.0448·29-s − 1.58·31-s + 2.46·33-s + 0.193·35-s + 0.0177·37-s − 1.58·39-s + 0.468·41-s + 0.539·43-s + 3.57·45-s + 0.232·47-s − 0.988·49-s − 1.23·51-s − 0.222·53-s + 2.54·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(6.698837248\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.698837248\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 + p T \) |
| good | 3 | \( 1 - p^{2} T + p^{3} T^{2} \) |
| 5 | \( 1 - 4 p T + p^{3} T^{2} \) |
| 7 | \( 1 - 2 T + p^{3} T^{2} \) |
| 11 | \( 1 - 52 T + p^{3} T^{2} \) |
| 13 | \( 1 + 43 T + p^{3} T^{2} \) |
| 17 | \( 1 + 50 T + p^{3} T^{2} \) |
| 19 | \( 1 - 74 T + p^{3} T^{2} \) |
| 29 | \( 1 - 7 T + p^{3} T^{2} \) |
| 31 | \( 1 + 273 T + p^{3} T^{2} \) |
| 37 | \( 1 - 4 T + p^{3} T^{2} \) |
| 41 | \( 1 - 3 p T + p^{3} T^{2} \) |
| 43 | \( 1 - 152 T + p^{3} T^{2} \) |
| 47 | \( 1 - 75 T + p^{3} T^{2} \) |
| 53 | \( 1 + 86 T + p^{3} T^{2} \) |
| 59 | \( 1 - 444 T + p^{3} T^{2} \) |
| 61 | \( 1 + 262 T + p^{3} T^{2} \) |
| 67 | \( 1 + 764 T + p^{3} T^{2} \) |
| 71 | \( 1 + 21 T + p^{3} T^{2} \) |
| 73 | \( 1 - 681 T + p^{3} T^{2} \) |
| 79 | \( 1 - 426 T + p^{3} T^{2} \) |
| 83 | \( 1 + 902 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1272 T + p^{3} T^{2} \) |
| 97 | \( 1 + 342 T + p^{3} T^{2} \) |
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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
−9.243439373511605307450673891777, −8.684543560371272273540195706014, −7.51501188274117799714930098266, −6.86605868644119456590286855080, −5.94179455336133752149891559931, −4.86785875036804402741617153646, −3.82673191127467633470182714134, −2.80505058016667897649683502519, −2.03265027072963140958805296244, −1.37240600570326253455645671063,
1.37240600570326253455645671063, 2.03265027072963140958805296244, 2.80505058016667897649683502519, 3.82673191127467633470182714134, 4.86785875036804402741617153646, 5.94179455336133752149891559931, 6.86605868644119456590286855080, 7.51501188274117799714930098266, 8.684543560371272273540195706014, 9.243439373511605307450673891777
