| L(s) = 1 | − 0.152·2-s − 3-s − 1.97·4-s − 5-s + 0.152·6-s − 2.73·7-s + 0.606·8-s + 9-s + 0.152·10-s − 0.976·11-s + 1.97·12-s − 2.15·13-s + 0.417·14-s + 15-s + 3.86·16-s − 5.31·17-s − 0.152·18-s + 7.23·19-s + 1.97·20-s + 2.73·21-s + 0.148·22-s + 5.21·23-s − 0.606·24-s + 25-s + 0.328·26-s − 27-s + 5.40·28-s + ⋯ |
| L(s) = 1 | − 0.107·2-s − 0.577·3-s − 0.988·4-s − 0.447·5-s + 0.0622·6-s − 1.03·7-s + 0.214·8-s + 0.333·9-s + 0.0482·10-s − 0.294·11-s + 0.570·12-s − 0.596·13-s + 0.111·14-s + 0.258·15-s + 0.965·16-s − 1.28·17-s − 0.0359·18-s + 1.65·19-s + 0.442·20-s + 0.596·21-s + 0.0317·22-s + 1.08·23-s − 0.123·24-s + 0.200·25-s + 0.0643·26-s − 0.192·27-s + 1.02·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 705 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 705 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5758844433\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5758844433\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 47 | \( 1 - T \) |
| good | 2 | \( 1 + 0.152T + 2T^{2} \) |
| 7 | \( 1 + 2.73T + 7T^{2} \) |
| 11 | \( 1 + 0.976T + 11T^{2} \) |
| 13 | \( 1 + 2.15T + 13T^{2} \) |
| 17 | \( 1 + 5.31T + 17T^{2} \) |
| 19 | \( 1 - 7.23T + 19T^{2} \) |
| 23 | \( 1 - 5.21T + 23T^{2} \) |
| 29 | \( 1 - 5T + 29T^{2} \) |
| 31 | \( 1 + 10.0T + 31T^{2} \) |
| 37 | \( 1 - 5.83T + 37T^{2} \) |
| 41 | \( 1 - 7.97T + 41T^{2} \) |
| 43 | \( 1 - 9.42T + 43T^{2} \) |
| 53 | \( 1 - 8.86T + 53T^{2} \) |
| 59 | \( 1 - 5.88T + 59T^{2} \) |
| 61 | \( 1 + 11.2T + 61T^{2} \) |
| 67 | \( 1 + 10.3T + 67T^{2} \) |
| 71 | \( 1 - 14.3T + 71T^{2} \) |
| 73 | \( 1 + 12.4T + 73T^{2} \) |
| 79 | \( 1 - 14.7T + 79T^{2} \) |
| 83 | \( 1 + 9.03T + 83T^{2} \) |
| 89 | \( 1 - 6.78T + 89T^{2} \) |
| 97 | \( 1 + 6.55T + 97T^{2} \) |
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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
−10.41193806406197431962991570592, −9.377530553699065195342658957285, −9.094325954254508951513559316430, −7.70196046281289215071171560408, −7.04238113089590116829006909756, −5.85056887388041110113384980556, −4.96656706775210278815858309841, −4.05923055466294356652616400216, −2.90428787350610435929061426855, −0.65866912698569379408310025305,
0.65866912698569379408310025305, 2.90428787350610435929061426855, 4.05923055466294356652616400216, 4.96656706775210278815858309841, 5.85056887388041110113384980556, 7.04238113089590116829006909756, 7.70196046281289215071171560408, 9.094325954254508951513559316430, 9.377530553699065195342658957285, 10.41193806406197431962991570592
