| L(s) = 1 | − 0.422·3-s + 5-s − 2.22·7-s − 2.82·9-s − 3.70·11-s − 0.141·13-s − 0.422·15-s − 6.15·17-s + 2.38·19-s + 0.939·21-s − 0.264·23-s + 25-s + 2.45·27-s + 0.575·29-s + 7.70·31-s + 1.56·33-s − 2.22·35-s − 2.83·37-s + 0.0599·39-s + 2.54·41-s − 11.1·43-s − 2.82·45-s − 5.40·47-s − 2.05·49-s + 2.59·51-s − 6.57·53-s − 3.70·55-s + ⋯ |
| L(s) = 1 | − 0.243·3-s + 0.447·5-s − 0.840·7-s − 0.940·9-s − 1.11·11-s − 0.0393·13-s − 0.109·15-s − 1.49·17-s + 0.548·19-s + 0.204·21-s − 0.0550·23-s + 0.200·25-s + 0.473·27-s + 0.106·29-s + 1.38·31-s + 0.272·33-s − 0.375·35-s − 0.465·37-s + 0.00959·39-s + 0.397·41-s − 1.70·43-s − 0.420·45-s − 0.788·47-s − 0.293·49-s + 0.363·51-s − 0.903·53-s − 0.499·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8554970048\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8554970048\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 151 | \( 1 + T \) |
| good | 3 | \( 1 + 0.422T + 3T^{2} \) |
| 7 | \( 1 + 2.22T + 7T^{2} \) |
| 11 | \( 1 + 3.70T + 11T^{2} \) |
| 13 | \( 1 + 0.141T + 13T^{2} \) |
| 17 | \( 1 + 6.15T + 17T^{2} \) |
| 19 | \( 1 - 2.38T + 19T^{2} \) |
| 23 | \( 1 + 0.264T + 23T^{2} \) |
| 29 | \( 1 - 0.575T + 29T^{2} \) |
| 31 | \( 1 - 7.70T + 31T^{2} \) |
| 37 | \( 1 + 2.83T + 37T^{2} \) |
| 41 | \( 1 - 2.54T + 41T^{2} \) |
| 43 | \( 1 + 11.1T + 43T^{2} \) |
| 47 | \( 1 + 5.40T + 47T^{2} \) |
| 53 | \( 1 + 6.57T + 53T^{2} \) |
| 59 | \( 1 - 3.40T + 59T^{2} \) |
| 61 | \( 1 + 2.76T + 61T^{2} \) |
| 67 | \( 1 - 12.7T + 67T^{2} \) |
| 71 | \( 1 - 8.14T + 71T^{2} \) |
| 73 | \( 1 + 6.38T + 73T^{2} \) |
| 79 | \( 1 + 10.7T + 79T^{2} \) |
| 83 | \( 1 - 6.69T + 83T^{2} \) |
| 89 | \( 1 - 5.81T + 89T^{2} \) |
| 97 | \( 1 - 8.17T + 97T^{2} \) |
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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
−8.275625960298895830381702909683, −7.25904555344540171023299226203, −6.45989696004627142947848459395, −6.10973928618052811588813846253, −5.14904260806573227655008177939, −4.71419067966883934456012517657, −3.40576686210881504153394807334, −2.81224673454795631364405992536, −2.01871003923426226083684983915, −0.46000721835540637076595618042,
0.46000721835540637076595618042, 2.01871003923426226083684983915, 2.81224673454795631364405992536, 3.40576686210881504153394807334, 4.71419067966883934456012517657, 5.14904260806573227655008177939, 6.10973928618052811588813846253, 6.45989696004627142947848459395, 7.25904555344540171023299226203, 8.275625960298895830381702909683
