| L(s) = 1 | + 0.189·3-s − 1.65·7-s − 2.96·9-s + 0.0759·11-s − 1.24·13-s + 5.17·17-s + 0.792·19-s − 0.313·21-s − 23-s − 1.12·27-s − 2.85·29-s + 8.90·31-s + 0.0143·33-s + 6.92·37-s − 0.235·39-s + 4.64·41-s − 1.75·43-s + 10.0·47-s − 4.25·49-s + 0.980·51-s − 11.1·53-s + 0.150·57-s − 12.4·59-s − 12.0·61-s + 4.91·63-s − 6.96·67-s − 0.189·69-s + ⋯ |
| L(s) = 1 | + 0.109·3-s − 0.626·7-s − 0.988·9-s + 0.0228·11-s − 0.344·13-s + 1.25·17-s + 0.181·19-s − 0.0684·21-s − 0.208·23-s − 0.217·27-s − 0.529·29-s + 1.59·31-s + 0.00250·33-s + 1.13·37-s − 0.0376·39-s + 0.725·41-s − 0.267·43-s + 1.45·47-s − 0.607·49-s + 0.137·51-s − 1.52·53-s + 0.0198·57-s − 1.62·59-s − 1.53·61-s + 0.618·63-s − 0.850·67-s − 0.0227·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 - 0.189T + 3T^{2} \) |
| 7 | \( 1 + 1.65T + 7T^{2} \) |
| 11 | \( 1 - 0.0759T + 11T^{2} \) |
| 13 | \( 1 + 1.24T + 13T^{2} \) |
| 17 | \( 1 - 5.17T + 17T^{2} \) |
| 19 | \( 1 - 0.792T + 19T^{2} \) |
| 29 | \( 1 + 2.85T + 29T^{2} \) |
| 31 | \( 1 - 8.90T + 31T^{2} \) |
| 37 | \( 1 - 6.92T + 37T^{2} \) |
| 41 | \( 1 - 4.64T + 41T^{2} \) |
| 43 | \( 1 + 1.75T + 43T^{2} \) |
| 47 | \( 1 - 10.0T + 47T^{2} \) |
| 53 | \( 1 + 11.1T + 53T^{2} \) |
| 59 | \( 1 + 12.4T + 59T^{2} \) |
| 61 | \( 1 + 12.0T + 61T^{2} \) |
| 67 | \( 1 + 6.96T + 67T^{2} \) |
| 71 | \( 1 + 7.71T + 71T^{2} \) |
| 73 | \( 1 + 7.49T + 73T^{2} \) |
| 79 | \( 1 + 15.6T + 79T^{2} \) |
| 83 | \( 1 - 6.68T + 83T^{2} \) |
| 89 | \( 1 + 3.03T + 89T^{2} \) |
| 97 | \( 1 + 3.21T + 97T^{2} \) |
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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
−7.80816151832521605201681421906, −7.48677221916982448718102501076, −6.17248247325326419853328191407, −6.03900913795832432923994858754, −5.02308577867296045845437443497, −4.16839834395885472374634820473, −3.08080063616651354388454507220, −2.75457097100686176368292047757, −1.32324582688478178892996734115, 0,
1.32324582688478178892996734115, 2.75457097100686176368292047757, 3.08080063616651354388454507220, 4.16839834395885472374634820473, 5.02308577867296045845437443497, 6.03900913795832432923994858754, 6.17248247325326419853328191407, 7.48677221916982448718102501076, 7.80816151832521605201681421906
