| L(s) = 1 | + 0.645·2-s + 2.49·3-s − 1.58·4-s + 1.61·6-s + 3.74·7-s − 2.31·8-s + 3.23·9-s + 6.02·11-s − 3.95·12-s + 3.48·13-s + 2.41·14-s + 1.67·16-s + 2.41·17-s + 2.09·18-s + 4.29·19-s + 9.34·21-s + 3.89·22-s + 7.88·23-s − 5.77·24-s + 2.24·26-s + 0.598·27-s − 5.92·28-s − 10.1·29-s − 5.58·31-s + 5.70·32-s + 15.0·33-s + 1.55·34-s + ⋯ |
| L(s) = 1 | + 0.456·2-s + 1.44·3-s − 0.791·4-s + 0.657·6-s + 1.41·7-s − 0.817·8-s + 1.07·9-s + 1.81·11-s − 1.14·12-s + 0.967·13-s + 0.645·14-s + 0.418·16-s + 0.585·17-s + 0.492·18-s + 0.985·19-s + 2.03·21-s + 0.829·22-s + 1.64·23-s − 1.17·24-s + 0.441·26-s + 0.115·27-s − 1.11·28-s − 1.87·29-s − 1.00·31-s + 1.00·32-s + 2.62·33-s + 0.267·34-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
| \(L(1)\) |
\(\approx\) |
\(5.130774288\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.130774288\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.645T + 2T^{2} \) |
| 3 | \( 1 - 2.49T + 3T^{2} \) |
| 7 | \( 1 - 3.74T + 7T^{2} \) |
| 11 | \( 1 - 6.02T + 11T^{2} \) |
| 13 | \( 1 - 3.48T + 13T^{2} \) |
| 17 | \( 1 - 2.41T + 17T^{2} \) |
| 19 | \( 1 - 4.29T + 19T^{2} \) |
| 23 | \( 1 - 7.88T + 23T^{2} \) |
| 29 | \( 1 + 10.1T + 29T^{2} \) |
| 31 | \( 1 + 5.58T + 31T^{2} \) |
| 37 | \( 1 + 2.37T + 37T^{2} \) |
| 41 | \( 1 + 1.48T + 41T^{2} \) |
| 43 | \( 1 + 1.44T + 43T^{2} \) |
| 47 | \( 1 + 7.27T + 47T^{2} \) |
| 53 | \( 1 + 9.63T + 53T^{2} \) |
| 59 | \( 1 + 2.82T + 59T^{2} \) |
| 61 | \( 1 - 3.52T + 61T^{2} \) |
| 67 | \( 1 + 6.26T + 67T^{2} \) |
| 71 | \( 1 - 7.39T + 71T^{2} \) |
| 73 | \( 1 + 3.34T + 73T^{2} \) |
| 79 | \( 1 + 4.80T + 79T^{2} \) |
| 83 | \( 1 - 5.42T + 83T^{2} \) |
| 89 | \( 1 - 7.49T + 89T^{2} \) |
| 97 | \( 1 + 12.1T + 97T^{2} \) |
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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
−8.191867001191369262506909260248, −7.59998586564231937299197573978, −6.80208227556979552758512459966, −5.71085385326812822113656173098, −5.07706457420564709627501704112, −4.23171419128893580100934488643, −3.55850057784934218914301959130, −3.22011408086851158352998484365, −1.70824695486579234051870495922, −1.25475719769851002536539006818,
1.25475719769851002536539006818, 1.70824695486579234051870495922, 3.22011408086851158352998484365, 3.55850057784934218914301959130, 4.23171419128893580100934488643, 5.07706457420564709627501704112, 5.71085385326812822113656173098, 6.80208227556979552758512459966, 7.59998586564231937299197573978, 8.191867001191369262506909260248
