| L(s) = 1 | − 3-s + 3.01·5-s + 3.11·7-s + 9-s + 4.01·11-s + 2.08·13-s − 3.01·15-s + 1.51·17-s − 3.39·19-s − 3.11·21-s + 23-s + 4.10·25-s − 27-s + 29-s − 0.311·31-s − 4.01·33-s + 9.40·35-s − 0.640·37-s − 2.08·39-s + 4.74·41-s + 2.47·43-s + 3.01·45-s − 8.97·47-s + 2.71·49-s − 1.51·51-s + 5.80·53-s + 12.1·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.34·5-s + 1.17·7-s + 0.333·9-s + 1.21·11-s + 0.577·13-s − 0.779·15-s + 0.366·17-s − 0.779·19-s − 0.680·21-s + 0.208·23-s + 0.821·25-s − 0.192·27-s + 0.185·29-s − 0.0560·31-s − 0.698·33-s + 1.58·35-s − 0.105·37-s − 0.333·39-s + 0.740·41-s + 0.378·43-s + 0.449·45-s − 1.30·47-s + 0.387·49-s − 0.211·51-s + 0.796·53-s + 1.63·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.174219891\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.174219891\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 5 | \( 1 - 3.01T + 5T^{2} \) |
| 7 | \( 1 - 3.11T + 7T^{2} \) |
| 11 | \( 1 - 4.01T + 11T^{2} \) |
| 13 | \( 1 - 2.08T + 13T^{2} \) |
| 17 | \( 1 - 1.51T + 17T^{2} \) |
| 19 | \( 1 + 3.39T + 19T^{2} \) |
| 31 | \( 1 + 0.311T + 31T^{2} \) |
| 37 | \( 1 + 0.640T + 37T^{2} \) |
| 41 | \( 1 - 4.74T + 41T^{2} \) |
| 43 | \( 1 - 2.47T + 43T^{2} \) |
| 47 | \( 1 + 8.97T + 47T^{2} \) |
| 53 | \( 1 - 5.80T + 53T^{2} \) |
| 59 | \( 1 + 11.3T + 59T^{2} \) |
| 61 | \( 1 - 10.1T + 61T^{2} \) |
| 67 | \( 1 - 9.23T + 67T^{2} \) |
| 71 | \( 1 - 10.8T + 71T^{2} \) |
| 73 | \( 1 - 13.2T + 73T^{2} \) |
| 79 | \( 1 - 15.2T + 79T^{2} \) |
| 83 | \( 1 + 7.22T + 83T^{2} \) |
| 89 | \( 1 + 8.32T + 89T^{2} \) |
| 97 | \( 1 + 19.6T + 97T^{2} \) |
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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
−7.955051723623462216443719673207, −6.79787641265832888873712509879, −6.49191450111284514557505260826, −5.70070442550565981498509126532, −5.20356360905151701838831304109, −4.40974287650529822894687552897, −3.67341833037723527826723712487, −2.37091254364068793935545852192, −1.64214973986750483602864610259, −1.01914591021815005318196023993,
1.01914591021815005318196023993, 1.64214973986750483602864610259, 2.37091254364068793935545852192, 3.67341833037723527826723712487, 4.40974287650529822894687552897, 5.20356360905151701838831304109, 5.70070442550565981498509126532, 6.49191450111284514557505260826, 6.79787641265832888873712509879, 7.955051723623462216443719673207
