| L(s) = 1 | − 2·4-s − 3.60·5-s − 7-s + 2·13-s + 4·16-s − 3.60·17-s + 6·19-s + 7.21·20-s − 7.21·23-s + 7.99·25-s + 2·28-s − 7.21·29-s − 2·31-s + 3.60·35-s − 2·37-s − 7.21·41-s − 5·43-s − 3.60·47-s + 49-s − 4·52-s + 10.8·59-s − 14·61-s − 8·64-s − 7.21·65-s − 15·67-s + 7.21·68-s + 14.4·71-s + ⋯ |
| L(s) = 1 | − 4-s − 1.61·5-s − 0.377·7-s + 0.554·13-s + 16-s − 0.874·17-s + 1.37·19-s + 1.61·20-s − 1.50·23-s + 1.59·25-s + 0.377·28-s − 1.33·29-s − 0.359·31-s + 0.609·35-s − 0.328·37-s − 1.12·41-s − 0.762·43-s − 0.525·47-s + 0.142·49-s − 0.554·52-s + 1.40·59-s − 1.79·61-s − 64-s − 0.894·65-s − 1.83·67-s + 0.874·68-s + 1.71·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3129689426\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3129689426\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 2T^{2} \) |
| 5 | \( 1 + 3.60T + 5T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 3.60T + 17T^{2} \) |
| 19 | \( 1 - 6T + 19T^{2} \) |
| 23 | \( 1 + 7.21T + 23T^{2} \) |
| 29 | \( 1 + 7.21T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + 7.21T + 41T^{2} \) |
| 43 | \( 1 + 5T + 43T^{2} \) |
| 47 | \( 1 + 3.60T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 10.8T + 59T^{2} \) |
| 61 | \( 1 + 14T + 61T^{2} \) |
| 67 | \( 1 + 15T + 67T^{2} \) |
| 71 | \( 1 - 14.4T + 71T^{2} \) |
| 73 | \( 1 - 4T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 10.8T + 83T^{2} \) |
| 89 | \( 1 + 10.8T + 89T^{2} \) |
| 97 | \( 1 + 8T + 97T^{2} \) |
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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
−7.889524339748084049185707811661, −7.40568877423436488142847225560, −6.57806456725513692220328344547, −5.64805292500454685917730112201, −4.95549140889292769683627988048, −4.09940126885914148897543728995, −3.72680187164579365453997043342, −3.08703167092752746816489756702, −1.57515174888093237177128256419, −0.28689729723752649835622371154,
0.28689729723752649835622371154, 1.57515174888093237177128256419, 3.08703167092752746816489756702, 3.72680187164579365453997043342, 4.09940126885914148897543728995, 4.95549140889292769683627988048, 5.64805292500454685917730112201, 6.57806456725513692220328344547, 7.40568877423436488142847225560, 7.889524339748084049185707811661
