| L(s) = 1 | − 0.481·3-s + 5-s − 7-s − 2.76·9-s − 11-s − 2.86·13-s − 0.481·15-s − 4.09·17-s + 5.35·19-s + 0.481·21-s − 8.46·23-s + 25-s + 2.77·27-s − 0.387·29-s + 7.28·31-s + 0.481·33-s − 35-s − 3.76·37-s + 1.38·39-s − 1.28·41-s − 7.50·43-s − 2.76·45-s + 13.1·47-s + 49-s + 1.96·51-s − 8.85·53-s − 55-s + ⋯ |
| L(s) = 1 | − 0.277·3-s + 0.447·5-s − 0.377·7-s − 0.922·9-s − 0.301·11-s − 0.795·13-s − 0.124·15-s − 0.992·17-s + 1.22·19-s + 0.105·21-s − 1.76·23-s + 0.200·25-s + 0.534·27-s − 0.0720·29-s + 1.30·31-s + 0.0837·33-s − 0.169·35-s − 0.619·37-s + 0.221·39-s − 0.201·41-s − 1.14·43-s − 0.412·45-s + 1.92·47-s + 0.142·49-s + 0.275·51-s − 1.21·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.098381731\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.098381731\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 + 0.481T + 3T^{2} \) |
| 13 | \( 1 + 2.86T + 13T^{2} \) |
| 17 | \( 1 + 4.09T + 17T^{2} \) |
| 19 | \( 1 - 5.35T + 19T^{2} \) |
| 23 | \( 1 + 8.46T + 23T^{2} \) |
| 29 | \( 1 + 0.387T + 29T^{2} \) |
| 31 | \( 1 - 7.28T + 31T^{2} \) |
| 37 | \( 1 + 3.76T + 37T^{2} \) |
| 41 | \( 1 + 1.28T + 41T^{2} \) |
| 43 | \( 1 + 7.50T + 43T^{2} \) |
| 47 | \( 1 - 13.1T + 47T^{2} \) |
| 53 | \( 1 + 8.85T + 53T^{2} \) |
| 59 | \( 1 - 8.24T + 59T^{2} \) |
| 61 | \( 1 + 3.28T + 61T^{2} \) |
| 67 | \( 1 + 3.50T + 67T^{2} \) |
| 71 | \( 1 - 14.2T + 71T^{2} \) |
| 73 | \( 1 - 6.79T + 73T^{2} \) |
| 79 | \( 1 + 10.2T + 79T^{2} \) |
| 83 | \( 1 - 6.77T + 83T^{2} \) |
| 89 | \( 1 - 7.22T + 89T^{2} \) |
| 97 | \( 1 + 1.61T + 97T^{2} \) |
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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
−8.109390468272576882490745129563, −7.30408405928309870132225363970, −6.52894859526747951796464101108, −5.93966178778670592123176202285, −5.26522547180109285388601786852, −4.59377291086146991949441645635, −3.53767611109991135476879695970, −2.69172241372341170808880270331, −2.00536045342019341844208949719, −0.52660627755566654147193779716,
0.52660627755566654147193779716, 2.00536045342019341844208949719, 2.69172241372341170808880270331, 3.53767611109991135476879695970, 4.59377291086146991949441645635, 5.26522547180109285388601786852, 5.93966178778670592123176202285, 6.52894859526747951796464101108, 7.30408405928309870132225363970, 8.109390468272576882490745129563
