| L(s) = 1 | + 3.27·3-s − 5-s − 4.39·7-s + 7.75·9-s + 1.69·11-s − 1.18·13-s − 3.27·15-s − 5.70·19-s − 14.4·21-s − 2.36·23-s + 25-s + 15.6·27-s − 7.22·29-s − 5.03·31-s + 5.56·33-s + 4.39·35-s − 5.45·37-s − 3.90·39-s + 4.45·41-s + 3.94·43-s − 7.75·45-s − 7.43·47-s + 12.3·49-s + 6.57·53-s − 1.69·55-s − 18.7·57-s + 4.87·59-s + ⋯ |
| L(s) = 1 | + 1.89·3-s − 0.447·5-s − 1.66·7-s + 2.58·9-s + 0.511·11-s − 0.329·13-s − 0.846·15-s − 1.30·19-s − 3.14·21-s − 0.494·23-s + 0.200·25-s + 3.00·27-s − 1.34·29-s − 0.905·31-s + 0.969·33-s + 0.743·35-s − 0.897·37-s − 0.624·39-s + 0.695·41-s + 0.601·43-s − 1.15·45-s − 1.08·47-s + 1.76·49-s + 0.903·53-s − 0.228·55-s − 2.48·57-s + 0.634·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5780 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5780 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 - 3.27T + 3T^{2} \) |
| 7 | \( 1 + 4.39T + 7T^{2} \) |
| 11 | \( 1 - 1.69T + 11T^{2} \) |
| 13 | \( 1 + 1.18T + 13T^{2} \) |
| 19 | \( 1 + 5.70T + 19T^{2} \) |
| 23 | \( 1 + 2.36T + 23T^{2} \) |
| 29 | \( 1 + 7.22T + 29T^{2} \) |
| 31 | \( 1 + 5.03T + 31T^{2} \) |
| 37 | \( 1 + 5.45T + 37T^{2} \) |
| 41 | \( 1 - 4.45T + 41T^{2} \) |
| 43 | \( 1 - 3.94T + 43T^{2} \) |
| 47 | \( 1 + 7.43T + 47T^{2} \) |
| 53 | \( 1 - 6.57T + 53T^{2} \) |
| 59 | \( 1 - 4.87T + 59T^{2} \) |
| 61 | \( 1 + 3.62T + 61T^{2} \) |
| 67 | \( 1 + 7.73T + 67T^{2} \) |
| 71 | \( 1 + 12.9T + 71T^{2} \) |
| 73 | \( 1 + 9.83T + 73T^{2} \) |
| 79 | \( 1 - 13.9T + 79T^{2} \) |
| 83 | \( 1 + 7.75T + 83T^{2} \) |
| 89 | \( 1 + 14.3T + 89T^{2} \) |
| 97 | \( 1 - 4.40T + 97T^{2} \) |
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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
−7.77637767623713285703833692183, −7.15929559775725363881191729210, −6.66478937837336802514112517323, −5.74657261202589716485904503754, −4.31326090267372445899548133226, −3.88073818533862091089895134279, −3.25641116806537433791451111737, −2.54083731214888066345146713540, −1.68542512957642989326725017255, 0,
1.68542512957642989326725017255, 2.54083731214888066345146713540, 3.25641116806537433791451111737, 3.88073818533862091089895134279, 4.31326090267372445899548133226, 5.74657261202589716485904503754, 6.66478937837336802514112517323, 7.15929559775725363881191729210, 7.77637767623713285703833692183
