| L(s) = 1 | + 1.28·2-s + 0.126·3-s − 0.345·4-s + 0.162·6-s − 1.03·7-s − 3.01·8-s − 2.98·9-s + 0.227·11-s − 0.0435·12-s − 3.38·13-s − 1.32·14-s − 3.18·16-s − 7.12·17-s − 3.83·18-s + 3.40·19-s − 0.130·21-s + 0.293·22-s − 6.91·23-s − 0.380·24-s − 4.34·26-s − 0.755·27-s + 0.356·28-s + 0.569·29-s + 4.93·31-s + 1.93·32-s + 0.0287·33-s − 9.16·34-s + ⋯ |
| L(s) = 1 | + 0.909·2-s + 0.0728·3-s − 0.172·4-s + 0.0662·6-s − 0.389·7-s − 1.06·8-s − 0.994·9-s + 0.0687·11-s − 0.0125·12-s − 0.937·13-s − 0.354·14-s − 0.797·16-s − 1.72·17-s − 0.904·18-s + 0.780·19-s − 0.0284·21-s + 0.0625·22-s − 1.44·23-s − 0.0777·24-s − 0.852·26-s − 0.145·27-s + 0.0672·28-s + 0.105·29-s + 0.886·31-s + 0.341·32-s + 0.00500·33-s − 1.57·34-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
| \(L(1)\) |
\(\approx\) |
\(1.378936551\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.378936551\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.28T + 2T^{2} \) |
| 3 | \( 1 - 0.126T + 3T^{2} \) |
| 7 | \( 1 + 1.03T + 7T^{2} \) |
| 11 | \( 1 - 0.227T + 11T^{2} \) |
| 13 | \( 1 + 3.38T + 13T^{2} \) |
| 17 | \( 1 + 7.12T + 17T^{2} \) |
| 19 | \( 1 - 3.40T + 19T^{2} \) |
| 23 | \( 1 + 6.91T + 23T^{2} \) |
| 29 | \( 1 - 0.569T + 29T^{2} \) |
| 31 | \( 1 - 4.93T + 31T^{2} \) |
| 37 | \( 1 - 5.37T + 37T^{2} \) |
| 41 | \( 1 - 10.7T + 41T^{2} \) |
| 43 | \( 1 - 0.910T + 43T^{2} \) |
| 47 | \( 1 - 8.50T + 47T^{2} \) |
| 53 | \( 1 - 7.76T + 53T^{2} \) |
| 59 | \( 1 - 11.2T + 59T^{2} \) |
| 61 | \( 1 + 3.65T + 61T^{2} \) |
| 67 | \( 1 - 12.0T + 67T^{2} \) |
| 71 | \( 1 + 9.48T + 71T^{2} \) |
| 73 | \( 1 - 7.10T + 73T^{2} \) |
| 79 | \( 1 - 0.366T + 79T^{2} \) |
| 83 | \( 1 + 17.8T + 83T^{2} \) |
| 89 | \( 1 + 7.54T + 89T^{2} \) |
| 97 | \( 1 - 7.85T + 97T^{2} \) |
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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
−8.143428440481123408350879489663, −7.24721841762354553556253936294, −6.38565095051655275357303245564, −5.87949695375732592987709601796, −5.17843821974442518388008635343, −4.35727901118106054646438336033, −3.86500991176374422709242163041, −2.70288945399940508326418172493, −2.42609835019884216306503343455, −0.50076888313895986608905428779,
0.50076888313895986608905428779, 2.42609835019884216306503343455, 2.70288945399940508326418172493, 3.86500991176374422709242163041, 4.35727901118106054646438336033, 5.17843821974442518388008635343, 5.87949695375732592987709601796, 6.38565095051655275357303245564, 7.24721841762354553556253936294, 8.143428440481123408350879489663
