| L(s) = 1 | + 0.895·2-s − 3-s − 1.19·4-s − 0.895·6-s − 5.08·7-s − 2.86·8-s + 9-s + 2.64·11-s + 1.19·12-s − 2.13·13-s − 4.55·14-s − 0.167·16-s − 7.75·17-s + 0.895·18-s + 3.08·19-s + 5.08·21-s + 2.36·22-s + 6.14·23-s + 2.86·24-s − 1.90·26-s − 27-s + 6.09·28-s − 4.13·29-s − 2.74·31-s + 5.57·32-s − 2.64·33-s − 6.93·34-s + ⋯ |
| L(s) = 1 | + 0.633·2-s − 0.577·3-s − 0.599·4-s − 0.365·6-s − 1.92·7-s − 1.01·8-s + 0.333·9-s + 0.796·11-s + 0.345·12-s − 0.591·13-s − 1.21·14-s − 0.0419·16-s − 1.87·17-s + 0.211·18-s + 0.708·19-s + 1.11·21-s + 0.504·22-s + 1.28·23-s + 0.584·24-s − 0.374·26-s − 0.192·27-s + 1.15·28-s − 0.767·29-s − 0.492·31-s + 0.985·32-s − 0.460·33-s − 1.19·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8199252944\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8199252944\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 2 | \( 1 - 0.895T + 2T^{2} \) |
| 7 | \( 1 + 5.08T + 7T^{2} \) |
| 11 | \( 1 - 2.64T + 11T^{2} \) |
| 13 | \( 1 + 2.13T + 13T^{2} \) |
| 17 | \( 1 + 7.75T + 17T^{2} \) |
| 19 | \( 1 - 3.08T + 19T^{2} \) |
| 23 | \( 1 - 6.14T + 23T^{2} \) |
| 29 | \( 1 + 4.13T + 29T^{2} \) |
| 31 | \( 1 + 2.74T + 31T^{2} \) |
| 37 | \( 1 - 0.0157T + 37T^{2} \) |
| 41 | \( 1 - 3.72T + 41T^{2} \) |
| 43 | \( 1 - 3.81T + 43T^{2} \) |
| 47 | \( 1 - 0.897T + 47T^{2} \) |
| 53 | \( 1 - 9.26T + 53T^{2} \) |
| 59 | \( 1 + 11.0T + 59T^{2} \) |
| 61 | \( 1 - 6.38T + 61T^{2} \) |
| 67 | \( 1 - 5.54T + 67T^{2} \) |
| 71 | \( 1 + 0.0828T + 71T^{2} \) |
| 73 | \( 1 - 9.92T + 73T^{2} \) |
| 79 | \( 1 - 5.30T + 79T^{2} \) |
| 83 | \( 1 + 0.723T + 83T^{2} \) |
| 89 | \( 1 + 13.2T + 89T^{2} \) |
| 97 | \( 1 - 2.22T + 97T^{2} \) |
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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
−9.284949188005913758709929598669, −8.851402191350877571992605870275, −7.24532150200851487084376808722, −6.68409055454067686410961111560, −6.01565871640092219182593857053, −5.19218331730439177880434685831, −4.23582637165371694803512521036, −3.55672126869236180636419376308, −2.57652888348014883204305798164, −0.54965167300245221101128906305,
0.54965167300245221101128906305, 2.57652888348014883204305798164, 3.55672126869236180636419376308, 4.23582637165371694803512521036, 5.19218331730439177880434685831, 6.01565871640092219182593857053, 6.68409055454067686410961111560, 7.24532150200851487084376808722, 8.851402191350877571992605870275, 9.284949188005913758709929598669
