| L(s) = 1 | − 1.61·3-s + 4.23·7-s − 0.381·9-s − 6.23·11-s − 2.38·13-s − 2.23·17-s + 4.09·19-s − 6.85·21-s + 5.23·23-s + 5.47·27-s − 10.2·29-s − 5.85·31-s + 10.0·33-s + 1.23·37-s + 3.85·39-s − 3·41-s − 2.70·43-s − 6.32·47-s + 10.9·49-s + 3.61·51-s − 9.85·53-s − 6.61·57-s − 3.38·59-s − 12.5·61-s − 1.61·63-s + 2.61·67-s − 8.47·69-s + ⋯ |
| L(s) = 1 | − 0.934·3-s + 1.60·7-s − 0.127·9-s − 1.88·11-s − 0.660·13-s − 0.542·17-s + 0.938·19-s − 1.49·21-s + 1.09·23-s + 1.05·27-s − 1.90·29-s − 1.05·31-s + 1.75·33-s + 0.203·37-s + 0.617·39-s − 0.468·41-s − 0.412·43-s − 0.922·47-s + 1.56·49-s + 0.506·51-s − 1.35·53-s − 0.876·57-s − 0.440·59-s − 1.60·61-s − 0.203·63-s + 0.319·67-s − 1.01·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 3 | \( 1 + 1.61T + 3T^{2} \) |
| 7 | \( 1 - 4.23T + 7T^{2} \) |
| 11 | \( 1 + 6.23T + 11T^{2} \) |
| 13 | \( 1 + 2.38T + 13T^{2} \) |
| 17 | \( 1 + 2.23T + 17T^{2} \) |
| 19 | \( 1 - 4.09T + 19T^{2} \) |
| 23 | \( 1 - 5.23T + 23T^{2} \) |
| 29 | \( 1 + 10.2T + 29T^{2} \) |
| 31 | \( 1 + 5.85T + 31T^{2} \) |
| 37 | \( 1 - 1.23T + 37T^{2} \) |
| 41 | \( 1 + 3T + 41T^{2} \) |
| 43 | \( 1 + 2.70T + 43T^{2} \) |
| 47 | \( 1 + 6.32T + 47T^{2} \) |
| 53 | \( 1 + 9.85T + 53T^{2} \) |
| 59 | \( 1 + 3.38T + 59T^{2} \) |
| 61 | \( 1 + 12.5T + 61T^{2} \) |
| 67 | \( 1 - 2.61T + 67T^{2} \) |
| 71 | \( 1 - 0.236T + 71T^{2} \) |
| 73 | \( 1 + 6.14T + 73T^{2} \) |
| 79 | \( 1 + 1.76T + 79T^{2} \) |
| 83 | \( 1 + 6.94T + 83T^{2} \) |
| 89 | \( 1 - 0.472T + 89T^{2} \) |
| 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
−9.672026645616532847217447079965, −8.615833115458923462306888894511, −7.73165603000303655126269241245, −7.24875942709918688118790580397, −5.78615563171547549546507287114, −5.12418584782094004229320198989, −4.76709001574561642755850287422, −3.01078191926190841212033857723, −1.77080709581073333036126883534, 0,
1.77080709581073333036126883534, 3.01078191926190841212033857723, 4.76709001574561642755850287422, 5.12418584782094004229320198989, 5.78615563171547549546507287114, 7.24875942709918688118790580397, 7.73165603000303655126269241245, 8.615833115458923462306888894511, 9.672026645616532847217447079965
