| L(s) = 1 | + 1.41·2-s + 3-s − 5-s + 1.41·6-s − 2.82·8-s + 9-s − 1.41·10-s − 2·11-s + 13-s − 15-s − 4.00·16-s − 1.41·17-s + 1.41·18-s − 6.41·19-s − 2.82·22-s − 2.41·23-s − 2.82·24-s − 4·25-s + 1.41·26-s + 27-s − 7.24·29-s − 1.41·30-s + 2.41·31-s − 2·33-s − 2.00·34-s + ⋯ |
| L(s) = 1 | + 1.00·2-s + 0.577·3-s − 0.447·5-s + 0.577·6-s − 0.999·8-s + 0.333·9-s − 0.447·10-s − 0.603·11-s + 0.277·13-s − 0.258·15-s − 1.00·16-s − 0.342·17-s + 0.333·18-s − 1.47·19-s − 0.603·22-s − 0.503·23-s − 0.577·24-s − 0.800·25-s + 0.277·26-s + 0.192·27-s − 1.34·29-s − 0.258·30-s + 0.433·31-s − 0.348·33-s − 0.342·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 - 1.41T + 2T^{2} \) |
| 5 | \( 1 + T + 5T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 17 | \( 1 + 1.41T + 17T^{2} \) |
| 19 | \( 1 + 6.41T + 19T^{2} \) |
| 23 | \( 1 + 2.41T + 23T^{2} \) |
| 29 | \( 1 + 7.24T + 29T^{2} \) |
| 31 | \( 1 - 2.41T + 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 - 10.8T + 41T^{2} \) |
| 43 | \( 1 + 1.82T + 43T^{2} \) |
| 47 | \( 1 - 6.65T + 47T^{2} \) |
| 53 | \( 1 + 0.414T + 53T^{2} \) |
| 59 | \( 1 + 3.65T + 59T^{2} \) |
| 61 | \( 1 + 0.343T + 61T^{2} \) |
| 67 | \( 1 + 7.41T + 67T^{2} \) |
| 71 | \( 1 + 1.17T + 71T^{2} \) |
| 73 | \( 1 + 8.41T + 73T^{2} \) |
| 79 | \( 1 + 15.4T + 79T^{2} \) |
| 83 | \( 1 - 11.4T + 83T^{2} \) |
| 89 | \( 1 + 8.65T + 89T^{2} \) |
| 97 | \( 1 - 2.89T + 97T^{2} \) |
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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
−8.760251703819577128001627280330, −8.070618728059663424381349750674, −7.25436371632256678799256700772, −6.19412075463920484268310558303, −5.55394744710465133331229211670, −4.35935818842015462933099405280, −4.05169929909269610530377443167, −3.01667820728312016077012342912, −2.06783125106069819011629902465, 0,
2.06783125106069819011629902465, 3.01667820728312016077012342912, 4.05169929909269610530377443167, 4.35935818842015462933099405280, 5.55394744710465133331229211670, 6.19412075463920484268310558303, 7.25436371632256678799256700772, 8.070618728059663424381349750674, 8.760251703819577128001627280330
