| L(s) = 1 | + 2.15·2-s + 2.64·4-s − 3.62·5-s + 2.27·7-s + 1.39·8-s − 7.81·10-s − 1.23·13-s + 4.89·14-s − 2.28·16-s + 5.69·17-s + 2.89·19-s − 9.59·20-s − 5.91·23-s + 8.13·25-s − 2.66·26-s + 6.00·28-s − 3.50·29-s − 2.50·31-s − 7.72·32-s + 12.2·34-s − 8.22·35-s − 0.333·37-s + 6.23·38-s − 5.05·40-s − 9.96·41-s + 7.15·43-s − 12.7·46-s + ⋯ |
| L(s) = 1 | + 1.52·2-s + 1.32·4-s − 1.62·5-s + 0.858·7-s + 0.492·8-s − 2.47·10-s − 0.343·13-s + 1.30·14-s − 0.572·16-s + 1.38·17-s + 0.664·19-s − 2.14·20-s − 1.23·23-s + 1.62·25-s − 0.523·26-s + 1.13·28-s − 0.651·29-s − 0.450·31-s − 1.36·32-s + 2.10·34-s − 1.39·35-s − 0.0549·37-s + 1.01·38-s − 0.798·40-s − 1.55·41-s + 1.09·43-s − 1.88·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9801 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9801 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 2.15T + 2T^{2} \) |
| 5 | \( 1 + 3.62T + 5T^{2} \) |
| 7 | \( 1 - 2.27T + 7T^{2} \) |
| 13 | \( 1 + 1.23T + 13T^{2} \) |
| 17 | \( 1 - 5.69T + 17T^{2} \) |
| 19 | \( 1 - 2.89T + 19T^{2} \) |
| 23 | \( 1 + 5.91T + 23T^{2} \) |
| 29 | \( 1 + 3.50T + 29T^{2} \) |
| 31 | \( 1 + 2.50T + 31T^{2} \) |
| 37 | \( 1 + 0.333T + 37T^{2} \) |
| 41 | \( 1 + 9.96T + 41T^{2} \) |
| 43 | \( 1 - 7.15T + 43T^{2} \) |
| 47 | \( 1 - 8.69T + 47T^{2} \) |
| 53 | \( 1 + 6.16T + 53T^{2} \) |
| 59 | \( 1 + 2.91T + 59T^{2} \) |
| 61 | \( 1 + 6.27T + 61T^{2} \) |
| 67 | \( 1 - 9.36T + 67T^{2} \) |
| 71 | \( 1 - 12.1T + 71T^{2} \) |
| 73 | \( 1 + 4.31T + 73T^{2} \) |
| 79 | \( 1 + 1.41T + 79T^{2} \) |
| 83 | \( 1 + 2.75T + 83T^{2} \) |
| 89 | \( 1 + 4.77T + 89T^{2} \) |
| 97 | \( 1 + 2.55T + 97T^{2} \) |
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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
−7.39823935583508615723624956024, −6.58056117778037680818574512963, −5.60375171508665975048152698821, −5.20148823179599280433252233387, −4.46334222661666623401152448006, −3.84064457516873437510831114806, −3.43639903351649572315428882558, −2.54808373905126001997175929266, −1.40614317706109510971434320084, 0,
1.40614317706109510971434320084, 2.54808373905126001997175929266, 3.43639903351649572315428882558, 3.84064457516873437510831114806, 4.46334222661666623401152448006, 5.20148823179599280433252233387, 5.60375171508665975048152698821, 6.58056117778037680818574512963, 7.39823935583508615723624956024
