L(s) = 1 | + 3.08·3-s − 3.57·5-s − 1.44·7-s + 6.53·9-s + 5.34·11-s − 5.39·13-s − 11.0·15-s − 5.31·17-s − 2.56·19-s − 4.45·21-s + 6.63·23-s + 7.81·25-s + 10.9·27-s + 5.19·29-s − 4.77·31-s + 16.5·33-s + 5.16·35-s − 3.44·37-s − 16.6·39-s + 8.38·41-s − 5.49·43-s − 23.3·45-s + 0.427·47-s − 4.92·49-s − 16.4·51-s − 3.73·53-s − 19.1·55-s + ⋯ |
L(s) = 1 | + 1.78·3-s − 1.60·5-s − 0.545·7-s + 2.17·9-s + 1.61·11-s − 1.49·13-s − 2.85·15-s − 1.28·17-s − 0.588·19-s − 0.971·21-s + 1.38·23-s + 1.56·25-s + 2.09·27-s + 0.964·29-s − 0.856·31-s + 2.87·33-s + 0.872·35-s − 0.565·37-s − 2.66·39-s + 1.31·41-s − 0.838·43-s − 3.48·45-s + 0.0624·47-s − 0.702·49-s − 2.29·51-s − 0.512·53-s − 2.57·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8752 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8752 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 547 | \( 1 - T \) |
good | 3 | \( 1 - 3.08T + 3T^{2} \) |
| 5 | \( 1 + 3.57T + 5T^{2} \) |
| 7 | \( 1 + 1.44T + 7T^{2} \) |
| 11 | \( 1 - 5.34T + 11T^{2} \) |
| 13 | \( 1 + 5.39T + 13T^{2} \) |
| 17 | \( 1 + 5.31T + 17T^{2} \) |
| 19 | \( 1 + 2.56T + 19T^{2} \) |
| 23 | \( 1 - 6.63T + 23T^{2} \) |
| 29 | \( 1 - 5.19T + 29T^{2} \) |
| 31 | \( 1 + 4.77T + 31T^{2} \) |
| 37 | \( 1 + 3.44T + 37T^{2} \) |
| 41 | \( 1 - 8.38T + 41T^{2} \) |
| 43 | \( 1 + 5.49T + 43T^{2} \) |
| 47 | \( 1 - 0.427T + 47T^{2} \) |
| 53 | \( 1 + 3.73T + 53T^{2} \) |
| 59 | \( 1 + 2.87T + 59T^{2} \) |
| 61 | \( 1 + 3.71T + 61T^{2} \) |
| 67 | \( 1 + 6.57T + 67T^{2} \) |
| 71 | \( 1 - 11.1T + 71T^{2} \) |
| 73 | \( 1 + 0.468T + 73T^{2} \) |
| 79 | \( 1 + 5.80T + 79T^{2} \) |
| 83 | \( 1 + 0.338T + 83T^{2} \) |
| 89 | \( 1 - 1.79T + 89T^{2} \) |
| 97 | \( 1 + 11.5T + 97T^{2} \) |
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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
−7.47444541482648256951878621882, −6.88090823547864929039192398807, −6.62482811560642991457866125372, −4.87337808674028001862569667723, −4.29393745327069126132521616617, −3.82934793382740670265835742059, −3.08720739951735096917854006550, −2.51004642010571295499639497566, −1.40549578410315206530676123228, 0,
1.40549578410315206530676123228, 2.51004642010571295499639497566, 3.08720739951735096917854006550, 3.82934793382740670265835742059, 4.29393745327069126132521616617, 4.87337808674028001862569667723, 6.62482811560642991457866125372, 6.88090823547864929039192398807, 7.47444541482648256951878621882