| L(s) = 1 | − 1.65·3-s + 0.725·5-s − 1.29·7-s − 0.269·9-s + 3.71·11-s + 2.30·13-s − 1.19·15-s + 6.32·17-s − 0.0212·19-s + 2.14·21-s + 1.30·23-s − 4.47·25-s + 5.40·27-s − 2.06·29-s + 5.82·31-s − 6.14·33-s − 0.942·35-s − 7.25·37-s − 3.81·39-s + 8.77·41-s − 4.52·43-s − 0.195·45-s − 3.85·47-s − 5.31·49-s − 10.4·51-s − 6.86·53-s + 2.69·55-s + ⋯ |
| L(s) = 1 | − 0.954·3-s + 0.324·5-s − 0.491·7-s − 0.0896·9-s + 1.12·11-s + 0.640·13-s − 0.309·15-s + 1.53·17-s − 0.00487·19-s + 0.468·21-s + 0.272·23-s − 0.894·25-s + 1.03·27-s − 0.383·29-s + 1.04·31-s − 1.06·33-s − 0.159·35-s − 1.19·37-s − 0.610·39-s + 1.37·41-s − 0.689·43-s − 0.0290·45-s − 0.562·47-s − 0.758·49-s − 1.46·51-s − 0.943·53-s + 0.363·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.412096633\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.412096633\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 269 | \( 1 + T \) |
| good | 3 | \( 1 + 1.65T + 3T^{2} \) |
| 5 | \( 1 - 0.725T + 5T^{2} \) |
| 7 | \( 1 + 1.29T + 7T^{2} \) |
| 11 | \( 1 - 3.71T + 11T^{2} \) |
| 13 | \( 1 - 2.30T + 13T^{2} \) |
| 17 | \( 1 - 6.32T + 17T^{2} \) |
| 19 | \( 1 + 0.0212T + 19T^{2} \) |
| 23 | \( 1 - 1.30T + 23T^{2} \) |
| 29 | \( 1 + 2.06T + 29T^{2} \) |
| 31 | \( 1 - 5.82T + 31T^{2} \) |
| 37 | \( 1 + 7.25T + 37T^{2} \) |
| 41 | \( 1 - 8.77T + 41T^{2} \) |
| 43 | \( 1 + 4.52T + 43T^{2} \) |
| 47 | \( 1 + 3.85T + 47T^{2} \) |
| 53 | \( 1 + 6.86T + 53T^{2} \) |
| 59 | \( 1 + 7.84T + 59T^{2} \) |
| 61 | \( 1 - 3.84T + 61T^{2} \) |
| 67 | \( 1 - 9.31T + 67T^{2} \) |
| 71 | \( 1 - 0.643T + 71T^{2} \) |
| 73 | \( 1 + 2.97T + 73T^{2} \) |
| 79 | \( 1 + 0.418T + 79T^{2} \) |
| 83 | \( 1 - 15.6T + 83T^{2} \) |
| 89 | \( 1 - 10.3T + 89T^{2} \) |
| 97 | \( 1 - 5.07T + 97T^{2} \) |
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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
−8.372228525960342914380569649162, −7.61375687970555994711177676759, −6.55100430544147304714094861982, −6.25032696596805641788441595539, −5.57040258060339934528516231920, −4.82061949452151046088171028374, −3.73766046773862869153350459832, −3.11518416840174933991605960703, −1.69330086616727328574300114513, −0.73536141609062732824911942677,
0.73536141609062732824911942677, 1.69330086616727328574300114513, 3.11518416840174933991605960703, 3.73766046773862869153350459832, 4.82061949452151046088171028374, 5.57040258060339934528516231920, 6.25032696596805641788441595539, 6.55100430544147304714094861982, 7.61375687970555994711177676759, 8.372228525960342914380569649162
