L(s) = 1 | + 3-s + 7-s + 9-s + 5.41·11-s + 4.34·13-s + 1.07·17-s + 4.34·19-s + 21-s + 6.34·23-s + 27-s − 8.83·29-s − 4.34·31-s + 5.41·33-s − 8.68·37-s + 4.34·39-s + 8.34·41-s + 6.15·43-s − 6.83·47-s + 49-s + 1.07·51-s + 6.18·53-s + 4.34·57-s − 6.83·59-s − 4.52·61-s + 63-s + 6.34·69-s − 14.0·71-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.377·7-s + 0.333·9-s + 1.63·11-s + 1.20·13-s + 0.261·17-s + 0.995·19-s + 0.218·21-s + 1.32·23-s + 0.192·27-s − 1.64·29-s − 0.779·31-s + 0.943·33-s − 1.42·37-s + 0.694·39-s + 1.30·41-s + 0.938·43-s − 0.997·47-s + 0.142·49-s + 0.151·51-s + 0.849·53-s + 0.574·57-s − 0.890·59-s − 0.579·61-s + 0.125·63-s + 0.763·69-s − 1.67·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.218977227\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.218977227\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 11 | \( 1 - 5.41T + 11T^{2} \) |
| 13 | \( 1 - 4.34T + 13T^{2} \) |
| 17 | \( 1 - 1.07T + 17T^{2} \) |
| 19 | \( 1 - 4.34T + 19T^{2} \) |
| 23 | \( 1 - 6.34T + 23T^{2} \) |
| 29 | \( 1 + 8.83T + 29T^{2} \) |
| 31 | \( 1 + 4.34T + 31T^{2} \) |
| 37 | \( 1 + 8.68T + 37T^{2} \) |
| 41 | \( 1 - 8.34T + 41T^{2} \) |
| 43 | \( 1 - 6.15T + 43T^{2} \) |
| 47 | \( 1 + 6.83T + 47T^{2} \) |
| 53 | \( 1 - 6.18T + 53T^{2} \) |
| 59 | \( 1 + 6.83T + 59T^{2} \) |
| 61 | \( 1 + 4.52T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 14.0T + 71T^{2} \) |
| 73 | \( 1 + 11.1T + 73T^{2} \) |
| 79 | \( 1 - 0.680T + 79T^{2} \) |
| 83 | \( 1 + 6.83T + 83T^{2} \) |
| 89 | \( 1 - 6.49T + 89T^{2} \) |
| 97 | \( 1 + 10.4T + 97T^{2} \) |
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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
−8.672881964216600307944043228373, −7.49830807320932891030685331379, −7.20344651368272013663050767288, −6.17785235792307531220184315294, −5.53144955574465468595613735524, −4.47582151153130186384050806480, −3.68926053109828767532394821110, −3.16910756899205535055321063300, −1.75049036367339217763819388622, −1.13363329568442560291432284888,
1.13363329568442560291432284888, 1.75049036367339217763819388622, 3.16910756899205535055321063300, 3.68926053109828767532394821110, 4.47582151153130186384050806480, 5.53144955574465468595613735524, 6.17785235792307531220184315294, 7.20344651368272013663050767288, 7.49830807320932891030685331379, 8.672881964216600307944043228373