| L(s) = 1 | + 2-s − 2.99·3-s + 4-s − 2.87·5-s − 2.99·6-s − 2.15·7-s + 8-s + 5.95·9-s − 2.87·10-s + 2.44·11-s − 2.99·12-s − 1.94·13-s − 2.15·14-s + 8.61·15-s + 16-s − 0.897·17-s + 5.95·18-s − 6.18·19-s − 2.87·20-s + 6.45·21-s + 2.44·22-s + 23-s − 2.99·24-s + 3.29·25-s − 1.94·26-s − 8.85·27-s − 2.15·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.72·3-s + 0.5·4-s − 1.28·5-s − 1.22·6-s − 0.815·7-s + 0.353·8-s + 1.98·9-s − 0.910·10-s + 0.736·11-s − 0.864·12-s − 0.538·13-s − 0.576·14-s + 2.22·15-s + 0.250·16-s − 0.217·17-s + 1.40·18-s − 1.41·19-s − 0.643·20-s + 1.40·21-s + 0.520·22-s + 0.208·23-s − 0.610·24-s + 0.658·25-s − 0.380·26-s − 1.70·27-s − 0.407·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4429967421\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4429967421\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 131 | \( 1 - T \) |
| good | 3 | \( 1 + 2.99T + 3T^{2} \) |
| 5 | \( 1 + 2.87T + 5T^{2} \) |
| 7 | \( 1 + 2.15T + 7T^{2} \) |
| 11 | \( 1 - 2.44T + 11T^{2} \) |
| 13 | \( 1 + 1.94T + 13T^{2} \) |
| 17 | \( 1 + 0.897T + 17T^{2} \) |
| 19 | \( 1 + 6.18T + 19T^{2} \) |
| 29 | \( 1 - 6.77T + 29T^{2} \) |
| 31 | \( 1 + 7.19T + 31T^{2} \) |
| 37 | \( 1 + 8.79T + 37T^{2} \) |
| 41 | \( 1 - 7.80T + 41T^{2} \) |
| 43 | \( 1 + 8.84T + 43T^{2} \) |
| 47 | \( 1 + 5.02T + 47T^{2} \) |
| 53 | \( 1 + 0.432T + 53T^{2} \) |
| 59 | \( 1 + 3.44T + 59T^{2} \) |
| 61 | \( 1 + 4.80T + 61T^{2} \) |
| 67 | \( 1 + 2.46T + 67T^{2} \) |
| 71 | \( 1 + 7.62T + 71T^{2} \) |
| 73 | \( 1 + 11.2T + 73T^{2} \) |
| 79 | \( 1 - 6.83T + 79T^{2} \) |
| 83 | \( 1 - 11.3T + 83T^{2} \) |
| 89 | \( 1 - 8.32T + 89T^{2} \) |
| 97 | \( 1 - 8.84T + 97T^{2} \) |
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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
−7.75500868073534435313178133214, −6.95323040048974045868617789747, −6.60566492385595316882228656988, −6.03010558698891424969132234273, −5.09961825799527093912368288031, −4.50031265003808050158101795620, −3.94041801500058909817160763850, −3.12632660138399863789699432132, −1.69161307376404664086696623713, −0.33837964816206066382316130018,
0.33837964816206066382316130018, 1.69161307376404664086696623713, 3.12632660138399863789699432132, 3.94041801500058909817160763850, 4.50031265003808050158101795620, 5.09961825799527093912368288031, 6.03010558698891424969132234273, 6.60566492385595316882228656988, 6.95323040048974045868617789747, 7.75500868073534435313178133214
