| L(s) = 1 | + 3-s − 7-s + 9-s − 5·11-s + 4·13-s + 2·17-s + 4·19-s − 21-s + 9·23-s + 27-s − 5·29-s + 2·31-s − 5·33-s − 11·37-s + 4·39-s + 8·41-s − 11·43-s + 12·47-s + 49-s + 2·51-s − 10·53-s + 4·57-s + 12·59-s − 12·61-s − 63-s + 3·67-s + 9·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.377·7-s + 1/3·9-s − 1.50·11-s + 1.10·13-s + 0.485·17-s + 0.917·19-s − 0.218·21-s + 1.87·23-s + 0.192·27-s − 0.928·29-s + 0.359·31-s − 0.870·33-s − 1.80·37-s + 0.640·39-s + 1.24·41-s − 1.67·43-s + 1.75·47-s + 1/7·49-s + 0.280·51-s − 1.37·53-s + 0.529·57-s + 1.56·59-s − 1.53·61-s − 0.125·63-s + 0.366·67-s + 1.08·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.278606686\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.278606686\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 7 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−8.513027144263135150492242042548, −7.55734494671456201641053518289, −7.24055680239777901861786668032, −6.16582607334000177087324402336, −5.38521208804387624402352072954, −4.75393871623385835532328490867, −3.39690911114168575821429232254, −3.20634884290031372513279250440, −2.05324129853614801218761405887, −0.841559754830924744185099046922,
0.841559754830924744185099046922, 2.05324129853614801218761405887, 3.20634884290031372513279250440, 3.39690911114168575821429232254, 4.75393871623385835532328490867, 5.38521208804387624402352072954, 6.16582607334000177087324402336, 7.24055680239777901861786668032, 7.55734494671456201641053518289, 8.513027144263135150492242042548
