| L(s) = 1 | + 2.30·2-s + 3-s + 3.30·4-s + 2.30·6-s + 7-s + 3.00·8-s + 9-s − 3·11-s + 3.30·12-s − 2.60·13-s + 2.30·14-s + 0.302·16-s + 4.60·17-s + 2.30·18-s + 6.60·19-s + 21-s − 6.90·22-s − 6.21·23-s + 3.00·24-s − 6·26-s + 27-s + 3.30·28-s − 7.60·29-s − 7.21·31-s − 5.30·32-s − 3·33-s + 10.6·34-s + ⋯ |
| L(s) = 1 | + 1.62·2-s + 0.577·3-s + 1.65·4-s + 0.940·6-s + 0.377·7-s + 1.06·8-s + 0.333·9-s − 0.904·11-s + 0.953·12-s − 0.722·13-s + 0.615·14-s + 0.0756·16-s + 1.11·17-s + 0.542·18-s + 1.51·19-s + 0.218·21-s − 1.47·22-s − 1.29·23-s + 0.612·24-s − 1.17·26-s + 0.192·27-s + 0.624·28-s − 1.41·29-s − 1.29·31-s − 0.937·32-s − 0.522·33-s + 1.81·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.986117763\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.986117763\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 2 | \( 1 - 2.30T + 2T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 + 2.60T + 13T^{2} \) |
| 17 | \( 1 - 4.60T + 17T^{2} \) |
| 19 | \( 1 - 6.60T + 19T^{2} \) |
| 23 | \( 1 + 6.21T + 23T^{2} \) |
| 29 | \( 1 + 7.60T + 29T^{2} \) |
| 31 | \( 1 + 7.21T + 31T^{2} \) |
| 37 | \( 1 + 4.21T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 9.60T + 43T^{2} \) |
| 47 | \( 1 - 10.6T + 47T^{2} \) |
| 53 | \( 1 - 3.21T + 53T^{2} \) |
| 59 | \( 1 + 10.6T + 59T^{2} \) |
| 61 | \( 1 + 1.21T + 61T^{2} \) |
| 67 | \( 1 - 15.6T + 67T^{2} \) |
| 71 | \( 1 + 3T + 71T^{2} \) |
| 73 | \( 1 - 0.605T + 73T^{2} \) |
| 79 | \( 1 + 14.8T + 79T^{2} \) |
| 83 | \( 1 + 3.21T + 83T^{2} \) |
| 89 | \( 1 - 7.81T + 89T^{2} \) |
| 97 | \( 1 + 0.788T + 97T^{2} \) |
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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
−11.13629895267005065564102425115, −10.12083640608244062401637699011, −9.156535412481457729549113279571, −7.64943234558559963596296521171, −7.38547276644971907099816709763, −5.71280330899916557445708472936, −5.29088170755339090135652733733, −4.07103391006517987507989755553, −3.17267331612027992051974079195, −2.08974155264634278796657756686,
2.08974155264634278796657756686, 3.17267331612027992051974079195, 4.07103391006517987507989755553, 5.29088170755339090135652733733, 5.71280330899916557445708472936, 7.38547276644971907099816709763, 7.64943234558559963596296521171, 9.156535412481457729549113279571, 10.12083640608244062401637699011, 11.13629895267005065564102425115
