| L(s) = 1 | + 2.63·2-s + 4.95·4-s − 5-s − 5.03·7-s + 7.78·8-s − 2.63·10-s + 2.09·11-s − 13.2·14-s + 10.6·16-s − 3.94·17-s + 1.13·19-s − 4.95·20-s + 5.53·22-s + 9.28·23-s + 25-s − 24.9·28-s + 9.15·29-s − 1.79·31-s + 12.4·32-s − 10.4·34-s + 5.03·35-s + 2.75·37-s + 3.00·38-s − 7.78·40-s − 2.34·41-s − 0.995·43-s + 10.3·44-s + ⋯ |
| L(s) = 1 | + 1.86·2-s + 2.47·4-s − 0.447·5-s − 1.90·7-s + 2.75·8-s − 0.833·10-s + 0.632·11-s − 3.54·14-s + 2.65·16-s − 0.957·17-s + 0.261·19-s − 1.10·20-s + 1.17·22-s + 1.93·23-s + 0.200·25-s − 4.71·28-s + 1.69·29-s − 0.321·31-s + 2.19·32-s − 1.78·34-s + 0.851·35-s + 0.453·37-s + 0.487·38-s − 1.23·40-s − 0.366·41-s − 0.151·43-s + 1.56·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.592746535\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.592746535\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 2.63T + 2T^{2} \) |
| 7 | \( 1 + 5.03T + 7T^{2} \) |
| 11 | \( 1 - 2.09T + 11T^{2} \) |
| 17 | \( 1 + 3.94T + 17T^{2} \) |
| 19 | \( 1 - 1.13T + 19T^{2} \) |
| 23 | \( 1 - 9.28T + 23T^{2} \) |
| 29 | \( 1 - 9.15T + 29T^{2} \) |
| 31 | \( 1 + 1.79T + 31T^{2} \) |
| 37 | \( 1 - 2.75T + 37T^{2} \) |
| 41 | \( 1 + 2.34T + 41T^{2} \) |
| 43 | \( 1 + 0.995T + 43T^{2} \) |
| 47 | \( 1 - 7.60T + 47T^{2} \) |
| 53 | \( 1 - 1.07T + 53T^{2} \) |
| 59 | \( 1 - 4.06T + 59T^{2} \) |
| 61 | \( 1 - 5.60T + 61T^{2} \) |
| 67 | \( 1 + 4.39T + 67T^{2} \) |
| 71 | \( 1 - 1.65T + 71T^{2} \) |
| 73 | \( 1 - 10.4T + 73T^{2} \) |
| 79 | \( 1 + 7.78T + 79T^{2} \) |
| 83 | \( 1 + 7.35T + 83T^{2} \) |
| 89 | \( 1 - 10.3T + 89T^{2} \) |
| 97 | \( 1 + 17.4T + 97T^{2} \) |
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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
−7.22439938782501284090578684307, −6.93572566108526315582692719696, −6.42514510714658353701688907971, −5.80398836593994475369508060948, −4.91842450331077003709680750281, −4.27034438018953584231300173670, −3.54959222767749954199605370110, −3.03335684300689333350882828668, −2.40204290732539153681597865272, −0.881211011838357908820425001689,
0.881211011838357908820425001689, 2.40204290732539153681597865272, 3.03335684300689333350882828668, 3.54959222767749954199605370110, 4.27034438018953584231300173670, 4.91842450331077003709680750281, 5.80398836593994475369508060948, 6.42514510714658353701688907971, 6.93572566108526315582692719696, 7.22439938782501284090578684307
