| L(s) = 1 | − 2.29·3-s − 5-s − 4.66·7-s + 2.24·9-s − 5.94·13-s + 2.29·15-s + 3.30·17-s + 4.19·19-s + 10.6·21-s − 1.71·23-s + 25-s + 1.72·27-s + 7.76·29-s − 2.97·31-s + 4.66·35-s − 5.68·37-s + 13.6·39-s + 6.14·41-s + 4.42·43-s − 2.24·45-s + 4.13·47-s + 14.7·49-s − 7.56·51-s + 8.10·53-s − 9.60·57-s − 6.21·59-s + 1.52·61-s + ⋯ |
| L(s) = 1 | − 1.32·3-s − 0.447·5-s − 1.76·7-s + 0.749·9-s − 1.64·13-s + 0.591·15-s + 0.801·17-s + 0.961·19-s + 2.33·21-s − 0.356·23-s + 0.200·25-s + 0.331·27-s + 1.44·29-s − 0.533·31-s + 0.788·35-s − 0.934·37-s + 2.17·39-s + 0.960·41-s + 0.674·43-s − 0.335·45-s + 0.602·47-s + 2.10·49-s − 1.05·51-s + 1.11·53-s − 1.27·57-s − 0.809·59-s + 0.195·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 2.29T + 3T^{2} \) |
| 7 | \( 1 + 4.66T + 7T^{2} \) |
| 13 | \( 1 + 5.94T + 13T^{2} \) |
| 17 | \( 1 - 3.30T + 17T^{2} \) |
| 19 | \( 1 - 4.19T + 19T^{2} \) |
| 23 | \( 1 + 1.71T + 23T^{2} \) |
| 29 | \( 1 - 7.76T + 29T^{2} \) |
| 31 | \( 1 + 2.97T + 31T^{2} \) |
| 37 | \( 1 + 5.68T + 37T^{2} \) |
| 41 | \( 1 - 6.14T + 41T^{2} \) |
| 43 | \( 1 - 4.42T + 43T^{2} \) |
| 47 | \( 1 - 4.13T + 47T^{2} \) |
| 53 | \( 1 - 8.10T + 53T^{2} \) |
| 59 | \( 1 + 6.21T + 59T^{2} \) |
| 61 | \( 1 - 1.52T + 61T^{2} \) |
| 67 | \( 1 + 7.46T + 67T^{2} \) |
| 71 | \( 1 + 2.23T + 71T^{2} \) |
| 73 | \( 1 - 7.49T + 73T^{2} \) |
| 79 | \( 1 + 17.5T + 79T^{2} \) |
| 83 | \( 1 + 11.0T + 83T^{2} \) |
| 89 | \( 1 - 12.8T + 89T^{2} \) |
| 97 | \( 1 + 9.53T + 97T^{2} \) |
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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
−7.44102813632363469589043533361, −7.24654513935000370589390728459, −6.36858536718453053710804059713, −5.77309512136863723207672804532, −5.12522369394024784394860998637, −4.28740148629256773033488794618, −3.28588219436335419537645454153, −2.61406507583942826013794687612, −0.853971366402880237445344380379, 0,
0.853971366402880237445344380379, 2.61406507583942826013794687612, 3.28588219436335419537645454153, 4.28740148629256773033488794618, 5.12522369394024784394860998637, 5.77309512136863723207672804532, 6.36858536718453053710804059713, 7.24654513935000370589390728459, 7.44102813632363469589043533361
