| L(s) = 1 | − 32·7-s − 36·11-s + 10·13-s − 78·17-s + 140·19-s − 192·23-s − 6·29-s − 16·31-s + 34·37-s + 390·41-s + 52·43-s + 408·47-s + 681·49-s − 114·53-s − 516·59-s − 58·61-s + 892·67-s + 120·71-s + 646·73-s + 1.15e3·77-s − 1.16e3·79-s − 732·83-s + 1.59e3·89-s − 320·91-s − 194·97-s − 798·101-s − 272·103-s + ⋯ |
| L(s) = 1 | − 1.72·7-s − 0.986·11-s + 0.213·13-s − 1.11·17-s + 1.69·19-s − 1.74·23-s − 0.0384·29-s − 0.0926·31-s + 0.151·37-s + 1.48·41-s + 0.184·43-s + 1.26·47-s + 1.98·49-s − 0.295·53-s − 1.13·59-s − 0.121·61-s + 1.62·67-s + 0.200·71-s + 1.03·73-s + 1.70·77-s − 1.66·79-s − 0.968·83-s + 1.89·89-s − 0.368·91-s − 0.203·97-s − 0.786·101-s − 0.260·103-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
| \(L(2)\) |
\(\approx\) |
\(1.019860809\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.019860809\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 32 T + p^{3} T^{2} \) |
| 11 | \( 1 + 36 T + p^{3} T^{2} \) |
| 13 | \( 1 - 10 T + p^{3} T^{2} \) |
| 17 | \( 1 + 78 T + p^{3} T^{2} \) |
| 19 | \( 1 - 140 T + p^{3} T^{2} \) |
| 23 | \( 1 + 192 T + p^{3} T^{2} \) |
| 29 | \( 1 + 6 T + p^{3} T^{2} \) |
| 31 | \( 1 + 16 T + p^{3} T^{2} \) |
| 37 | \( 1 - 34 T + p^{3} T^{2} \) |
| 41 | \( 1 - 390 T + p^{3} T^{2} \) |
| 43 | \( 1 - 52 T + p^{3} T^{2} \) |
| 47 | \( 1 - 408 T + p^{3} T^{2} \) |
| 53 | \( 1 + 114 T + p^{3} T^{2} \) |
| 59 | \( 1 + 516 T + p^{3} T^{2} \) |
| 61 | \( 1 + 58 T + p^{3} T^{2} \) |
| 67 | \( 1 - 892 T + p^{3} T^{2} \) |
| 71 | \( 1 - 120 T + p^{3} T^{2} \) |
| 73 | \( 1 - 646 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1168 T + p^{3} T^{2} \) |
| 83 | \( 1 + 732 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1590 T + p^{3} T^{2} \) |
| 97 | \( 1 + 2 p T + p^{3} T^{2} \) |
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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
−9.700742514860869623340503536998, −9.112326393592408190633976905732, −7.946322946767269885294177224774, −7.16440043938530887919893927834, −6.20483375631328835507494774733, −5.56127738374691518801080865708, −4.22045135357860099982994653228, −3.23993841188587717907408376483, −2.34674579099382886257390226005, −0.52117229361164425806720285939,
0.52117229361164425806720285939, 2.34674579099382886257390226005, 3.23993841188587717907408376483, 4.22045135357860099982994653228, 5.56127738374691518801080865708, 6.20483375631328835507494774733, 7.16440043938530887919893927834, 7.946322946767269885294177224774, 9.112326393592408190633976905732, 9.700742514860869623340503536998
