| L(s) = 1 | − 1.73·3-s − 5.27·11-s − 2.62·13-s − 0.418·17-s + 3.27·19-s + 7.82·23-s + 5.19·27-s + 4.27·29-s + 3.27·31-s + 9.13·33-s + 9.97·37-s + 4.54·39-s − 3.72·41-s + 2.15·43-s − 6.50·47-s + 0.725·51-s + 5.67·53-s − 5.67·57-s − 3.27·59-s − 13.5·61-s − 3.52·67-s − 13.5·69-s − 4.54·71-s − 6.50·73-s + 7.27·79-s − 9·81-s + 7.40·83-s + ⋯ |
| L(s) = 1 | − 1.00·3-s − 1.59·11-s − 0.728·13-s − 0.101·17-s + 0.751·19-s + 1.63·23-s + 1.00·27-s + 0.793·29-s + 0.588·31-s + 1.59·33-s + 1.63·37-s + 0.728·39-s − 0.581·41-s + 0.327·43-s − 0.949·47-s + 0.101·51-s + 0.779·53-s − 0.751·57-s − 0.426·59-s − 1.73·61-s − 0.430·67-s − 1.63·69-s − 0.539·71-s − 0.761·73-s + 0.818·79-s − 81-s + 0.812·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 1.73T + 3T^{2} \) |
| 11 | \( 1 + 5.27T + 11T^{2} \) |
| 13 | \( 1 + 2.62T + 13T^{2} \) |
| 17 | \( 1 + 0.418T + 17T^{2} \) |
| 19 | \( 1 - 3.27T + 19T^{2} \) |
| 23 | \( 1 - 7.82T + 23T^{2} \) |
| 29 | \( 1 - 4.27T + 29T^{2} \) |
| 31 | \( 1 - 3.27T + 31T^{2} \) |
| 37 | \( 1 - 9.97T + 37T^{2} \) |
| 41 | \( 1 + 3.72T + 41T^{2} \) |
| 43 | \( 1 - 2.15T + 43T^{2} \) |
| 47 | \( 1 + 6.50T + 47T^{2} \) |
| 53 | \( 1 - 5.67T + 53T^{2} \) |
| 59 | \( 1 + 3.27T + 59T^{2} \) |
| 61 | \( 1 + 13.5T + 61T^{2} \) |
| 67 | \( 1 + 3.52T + 67T^{2} \) |
| 71 | \( 1 + 4.54T + 71T^{2} \) |
| 73 | \( 1 + 6.50T + 73T^{2} \) |
| 79 | \( 1 - 7.27T + 79T^{2} \) |
| 83 | \( 1 - 7.40T + 83T^{2} \) |
| 89 | \( 1 + 7T + 89T^{2} \) |
| 97 | \( 1 - 6.92T + 97T^{2} \) |
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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
−7.79981954324578547329765068861, −7.17175736037290507174760610905, −6.37048779003466986050488837543, −5.61742069232045284834503696962, −4.98944973476097996149396645089, −4.58421084460227091724048603117, −3.03505922042885497678002408675, −2.62814911109017780709419780341, −1.07342434896189540411707306242, 0,
1.07342434896189540411707306242, 2.62814911109017780709419780341, 3.03505922042885497678002408675, 4.58421084460227091724048603117, 4.98944973476097996149396645089, 5.61742069232045284834503696962, 6.37048779003466986050488837543, 7.17175736037290507174760610905, 7.79981954324578547329765068861
