| L(s) = 1 | + 2.37·2-s − 1.45·3-s + 3.63·4-s − 3.45·6-s − 1.94·7-s + 3.89·8-s − 0.887·9-s − 6.31·11-s − 5.28·12-s + 2.12·13-s − 4.62·14-s + 1.96·16-s + 0.676·17-s − 2.10·18-s + 6.14·19-s + 2.82·21-s − 15.0·22-s − 1.31·23-s − 5.65·24-s + 5.04·26-s + 5.65·27-s − 7.08·28-s + 7.00·29-s + 6.77·31-s − 3.11·32-s + 9.18·33-s + 1.60·34-s + ⋯ |
| L(s) = 1 | + 1.67·2-s − 0.839·3-s + 1.81·4-s − 1.40·6-s − 0.735·7-s + 1.37·8-s − 0.295·9-s − 1.90·11-s − 1.52·12-s + 0.589·13-s − 1.23·14-s + 0.491·16-s + 0.164·17-s − 0.496·18-s + 1.41·19-s + 0.617·21-s − 3.19·22-s − 0.274·23-s − 1.15·24-s + 0.990·26-s + 1.08·27-s − 1.33·28-s + 1.30·29-s + 1.21·31-s − 0.551·32-s + 1.59·33-s + 0.275·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.965339157\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.965339157\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 241 | \( 1 - T \) |
| good | 2 | \( 1 - 2.37T + 2T^{2} \) |
| 3 | \( 1 + 1.45T + 3T^{2} \) |
| 7 | \( 1 + 1.94T + 7T^{2} \) |
| 11 | \( 1 + 6.31T + 11T^{2} \) |
| 13 | \( 1 - 2.12T + 13T^{2} \) |
| 17 | \( 1 - 0.676T + 17T^{2} \) |
| 19 | \( 1 - 6.14T + 19T^{2} \) |
| 23 | \( 1 + 1.31T + 23T^{2} \) |
| 29 | \( 1 - 7.00T + 29T^{2} \) |
| 31 | \( 1 - 6.77T + 31T^{2} \) |
| 37 | \( 1 + 6.97T + 37T^{2} \) |
| 41 | \( 1 - 7.66T + 41T^{2} \) |
| 43 | \( 1 + 6.41T + 43T^{2} \) |
| 47 | \( 1 + 1.24T + 47T^{2} \) |
| 53 | \( 1 - 0.377T + 53T^{2} \) |
| 59 | \( 1 - 11.9T + 59T^{2} \) |
| 61 | \( 1 - 9.81T + 61T^{2} \) |
| 67 | \( 1 - 2.68T + 67T^{2} \) |
| 71 | \( 1 - 6.49T + 71T^{2} \) |
| 73 | \( 1 - 6.10T + 73T^{2} \) |
| 79 | \( 1 - 6.30T + 79T^{2} \) |
| 83 | \( 1 - 15.5T + 83T^{2} \) |
| 89 | \( 1 + 11.1T + 89T^{2} \) |
| 97 | \( 1 + 1.84T + 97T^{2} \) |
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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
−7.88615223719533765019260128600, −6.90506902151358580369897321968, −6.41415676946167644207676014590, −5.68082660064962041053366195295, −5.22369814393393650562039022980, −4.76936240919374716308781529892, −3.60753683316527560284557059975, −3.01535802239896414091234751529, −2.38339334713915115214453221377, −0.69864737432597796152134848950,
0.69864737432597796152134848950, 2.38339334713915115214453221377, 3.01535802239896414091234751529, 3.60753683316527560284557059975, 4.76936240919374716308781529892, 5.22369814393393650562039022980, 5.68082660064962041053366195295, 6.41415676946167644207676014590, 6.90506902151358580369897321968, 7.88615223719533765019260128600
