| L(s) = 1 | + 23.3·7-s − 63.2·11-s + 69.9·13-s + 92.1·17-s + 12·19-s − 184.·23-s + 189.·29-s + 136·31-s − 116.·37-s + 126.·41-s − 186.·43-s + 184.·47-s + 201·49-s + 276.·53-s − 316.·59-s + 794·61-s + 326.·67-s − 379.·71-s − 466.·73-s − 1.47e3·77-s + 384·79-s + 368.·83-s − 1.01e3·89-s + 1.63e3·91-s − 1.07e3·97-s + 1.58e3·101-s + 1.51e3·103-s + ⋯ |
| L(s) = 1 | + 1.25·7-s − 1.73·11-s + 1.49·13-s + 1.31·17-s + 0.144·19-s − 1.67·23-s + 1.21·29-s + 0.787·31-s − 0.518·37-s + 0.481·41-s − 0.661·43-s + 0.572·47-s + 0.586·49-s + 0.716·53-s − 0.697·59-s + 1.66·61-s + 0.595·67-s − 0.634·71-s − 0.747·73-s − 2.18·77-s + 0.546·79-s + 0.487·83-s − 1.20·89-s + 1.88·91-s − 1.12·97-s + 1.55·101-s + 1.45·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.427527715\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.427527715\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 23.3T + 343T^{2} \) |
| 11 | \( 1 + 63.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 69.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 92.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 12T + 6.85e3T^{2} \) |
| 23 | \( 1 + 184.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 189.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 136T + 2.97e4T^{2} \) |
| 37 | \( 1 + 116.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 126.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 186.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 184.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 276.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 316.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 794T + 2.26e5T^{2} \) |
| 67 | \( 1 - 326.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 379.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 466.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 384T + 4.93e5T^{2} \) |
| 83 | \( 1 - 368.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.01e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.07e3T + 9.12e5T^{2} \) |
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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
−10.04225217860977011660762793106, −8.451159115063872144892635472480, −8.215990046863907693745514729311, −7.43701325195995882282852520342, −6.05007735718348541258328084761, −5.38176371379443381716563312021, −4.45662749285768956841442508415, −3.28307656825421286126929857545, −2.06364130203291573707917918527, −0.882448921153928771942835260423,
0.882448921153928771942835260423, 2.06364130203291573707917918527, 3.28307656825421286126929857545, 4.45662749285768956841442508415, 5.38176371379443381716563312021, 6.05007735718348541258328084761, 7.43701325195995882282852520342, 8.215990046863907693745514729311, 8.451159115063872144892635472480, 10.04225217860977011660762793106
