| L(s) = 1 | + 5.36·3-s + 2.53·5-s − 32.2·7-s + 1.74·9-s − 6.42·11-s − 14.3·13-s + 13.5·15-s + 17·17-s + 72.4·19-s − 172.·21-s + 167.·23-s − 118.·25-s − 135.·27-s + 113.·29-s + 159.·31-s − 34.4·33-s − 81.5·35-s + 39.5·37-s − 76.9·39-s + 380.·41-s − 192.·43-s + 4.41·45-s + 309.·47-s + 695.·49-s + 91.1·51-s − 26.9·53-s − 16.2·55-s + ⋯ |
| L(s) = 1 | + 1.03·3-s + 0.226·5-s − 1.74·7-s + 0.0646·9-s − 0.176·11-s − 0.306·13-s + 0.233·15-s + 0.242·17-s + 0.874·19-s − 1.79·21-s + 1.51·23-s − 0.948·25-s − 0.965·27-s + 0.725·29-s + 0.924·31-s − 0.181·33-s − 0.393·35-s + 0.175·37-s − 0.316·39-s + 1.44·41-s − 0.684·43-s + 0.0146·45-s + 0.959·47-s + 2.02·49-s + 0.250·51-s − 0.0697·53-s − 0.0398·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.347555016\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.347555016\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 - 17T \) |
| good | 3 | \( 1 - 5.36T + 27T^{2} \) |
| 5 | \( 1 - 2.53T + 125T^{2} \) |
| 7 | \( 1 + 32.2T + 343T^{2} \) |
| 11 | \( 1 + 6.42T + 1.33e3T^{2} \) |
| 13 | \( 1 + 14.3T + 2.19e3T^{2} \) |
| 19 | \( 1 - 72.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 167.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 113.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 159.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 39.5T + 5.06e4T^{2} \) |
| 41 | \( 1 - 380.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 192.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 309.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 26.9T + 1.48e5T^{2} \) |
| 59 | \( 1 - 214.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 140.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 676.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 966.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 639.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.31e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.41e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 387.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 359.T + 9.12e5T^{2} \) |
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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
−9.593118205662487549582541339588, −8.819055252884168410731004230714, −7.87877921245511226552798456947, −7.04257256467414101401575771876, −6.20813362143413124560212639296, −5.26893841102423892791777112904, −3.86712358572640209380357394969, −3.04300378346977091631499910225, −2.48635773324160616198595331987, −0.74179410614538348667855869446,
0.74179410614538348667855869446, 2.48635773324160616198595331987, 3.04300378346977091631499910225, 3.86712358572640209380357394969, 5.26893841102423892791777112904, 6.20813362143413124560212639296, 7.04257256467414101401575771876, 7.87877921245511226552798456947, 8.819055252884168410731004230714, 9.593118205662487549582541339588
