| L(s) = 1 | + 3-s + 9-s − 2.86·11-s + 13-s − 5.52·17-s + 3.52·19-s − 7.52·23-s + 27-s + 6.77·29-s + 5.72·31-s − 2.86·33-s + 3.72·37-s + 39-s − 10.1·41-s + 5.52·43-s − 8.65·47-s − 7·49-s − 5.52·51-s + 6.77·53-s + 3.52·57-s + 0.593·59-s − 5.25·61-s + 10.5·67-s − 7.52·69-s − 2.38·71-s − 5.45·73-s + 2.47·79-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.333·9-s − 0.863·11-s + 0.277·13-s − 1.33·17-s + 0.808·19-s − 1.56·23-s + 0.192·27-s + 1.25·29-s + 1.02·31-s − 0.498·33-s + 0.613·37-s + 0.160·39-s − 1.58·41-s + 0.842·43-s − 1.26·47-s − 49-s − 0.773·51-s + 0.930·53-s + 0.466·57-s + 0.0773·59-s − 0.672·61-s + 1.28·67-s − 0.905·69-s − 0.283·71-s − 0.638·73-s + 0.278·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 2.86T + 11T^{2} \) |
| 17 | \( 1 + 5.52T + 17T^{2} \) |
| 19 | \( 1 - 3.52T + 19T^{2} \) |
| 23 | \( 1 + 7.52T + 23T^{2} \) |
| 29 | \( 1 - 6.77T + 29T^{2} \) |
| 31 | \( 1 - 5.72T + 31T^{2} \) |
| 37 | \( 1 - 3.72T + 37T^{2} \) |
| 41 | \( 1 + 10.1T + 41T^{2} \) |
| 43 | \( 1 - 5.52T + 43T^{2} \) |
| 47 | \( 1 + 8.65T + 47T^{2} \) |
| 53 | \( 1 - 6.77T + 53T^{2} \) |
| 59 | \( 1 - 0.593T + 59T^{2} \) |
| 61 | \( 1 + 5.25T + 61T^{2} \) |
| 67 | \( 1 - 10.5T + 67T^{2} \) |
| 71 | \( 1 + 2.38T + 71T^{2} \) |
| 73 | \( 1 + 5.45T + 73T^{2} \) |
| 79 | \( 1 - 2.47T + 79T^{2} \) |
| 83 | \( 1 + 8.11T + 83T^{2} \) |
| 89 | \( 1 + 14.1T + 89T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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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
−7.67215588937917696884675765736, −6.77953939702828842126594776782, −6.27699128499688535294274464680, −5.33471249725517386297250747019, −4.60479022217698224353097100293, −3.94130844640860387594086373671, −2.95804649553138258896601839949, −2.39642490723420324740181388671, −1.38887103708305694508247303011, 0,
1.38887103708305694508247303011, 2.39642490723420324740181388671, 2.95804649553138258896601839949, 3.94130844640860387594086373671, 4.60479022217698224353097100293, 5.33471249725517386297250747019, 6.27699128499688535294274464680, 6.77953939702828842126594776782, 7.67215588937917696884675765736
