| L(s) = 1 | − 2-s + 4-s + 1.07·7-s − 8-s − 6.34·11-s − 3.41·13-s − 1.07·14-s + 16-s − 5.41·17-s + 19-s + 6.34·22-s + 6.34·23-s + 3.41·26-s + 1.07·28-s + 0.340·29-s + 1.07·31-s − 32-s + 5.41·34-s − 3.41·37-s − 38-s − 7.60·41-s + 11.1·43-s − 6.34·44-s − 6.34·46-s − 6.34·47-s − 5.83·49-s − 3.41·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.407·7-s − 0.353·8-s − 1.91·11-s − 0.948·13-s − 0.288·14-s + 0.250·16-s − 1.31·17-s + 0.229·19-s + 1.35·22-s + 1.32·23-s + 0.670·26-s + 0.203·28-s + 0.0631·29-s + 0.193·31-s − 0.176·32-s + 0.929·34-s − 0.562·37-s − 0.162·38-s − 1.18·41-s + 1.70·43-s − 0.955·44-s − 0.934·46-s − 0.924·47-s − 0.833·49-s − 0.474·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7445075585\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7445075585\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 - 1.07T + 7T^{2} \) |
| 11 | \( 1 + 6.34T + 11T^{2} \) |
| 13 | \( 1 + 3.41T + 13T^{2} \) |
| 17 | \( 1 + 5.41T + 17T^{2} \) |
| 23 | \( 1 - 6.34T + 23T^{2} \) |
| 29 | \( 1 - 0.340T + 29T^{2} \) |
| 31 | \( 1 - 1.07T + 31T^{2} \) |
| 37 | \( 1 + 3.41T + 37T^{2} \) |
| 41 | \( 1 + 7.60T + 41T^{2} \) |
| 43 | \( 1 - 11.1T + 43T^{2} \) |
| 47 | \( 1 + 6.34T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + 0.738T + 59T^{2} \) |
| 61 | \( 1 + 2.68T + 61T^{2} \) |
| 67 | \( 1 + 2.83T + 67T^{2} \) |
| 71 | \( 1 - 2.83T + 71T^{2} \) |
| 73 | \( 1 + 6.83T + 73T^{2} \) |
| 79 | \( 1 + 1.07T + 79T^{2} \) |
| 83 | \( 1 - 0.894T + 83T^{2} \) |
| 89 | \( 1 + 6.92T + 89T^{2} \) |
| 97 | \( 1 - 3.65T + 97T^{2} \) |
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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
−7.77916823049008533927076673520, −7.24862451452729884646877695216, −6.67199170811667325021736370539, −5.63019648636041995016179304690, −5.02521117219958121031058577034, −4.48585639664261130236534176268, −3.11902141056990383847453744543, −2.56409794199333565931121708535, −1.79581902305663366305092530818, −0.44685535770629483631864430847,
0.44685535770629483631864430847, 1.79581902305663366305092530818, 2.56409794199333565931121708535, 3.11902141056990383847453744543, 4.48585639664261130236534176268, 5.02521117219958121031058577034, 5.63019648636041995016179304690, 6.67199170811667325021736370539, 7.24862451452729884646877695216, 7.77916823049008533927076673520
