| L(s) = 1 | − 0.927·2-s + 3-s − 1.13·4-s + 5-s − 0.927·6-s + 2.45·7-s + 2.91·8-s + 9-s − 0.927·10-s + 1.24·11-s − 1.13·12-s + 0.244·13-s − 2.27·14-s + 15-s − 0.423·16-s + 2.92·17-s − 0.927·18-s + 4.17·19-s − 1.13·20-s + 2.45·21-s − 1.15·22-s + 2.91·24-s + 25-s − 0.226·26-s + 27-s − 2.79·28-s + 2·29-s + ⋯ |
| L(s) = 1 | − 0.655·2-s + 0.577·3-s − 0.569·4-s + 0.447·5-s − 0.378·6-s + 0.928·7-s + 1.02·8-s + 0.333·9-s − 0.293·10-s + 0.375·11-s − 0.328·12-s + 0.0678·13-s − 0.609·14-s + 0.258·15-s − 0.105·16-s + 0.710·17-s − 0.218·18-s + 0.957·19-s − 0.254·20-s + 0.535·21-s − 0.246·22-s + 0.594·24-s + 0.200·25-s − 0.0445·26-s + 0.192·27-s − 0.528·28-s + 0.371·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.330497442\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.330497442\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 + 0.927T + 2T^{2} \) |
| 7 | \( 1 - 2.45T + 7T^{2} \) |
| 11 | \( 1 - 1.24T + 11T^{2} \) |
| 13 | \( 1 - 0.244T + 13T^{2} \) |
| 17 | \( 1 - 2.92T + 17T^{2} \) |
| 19 | \( 1 - 4.17T + 19T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 9.59T + 31T^{2} \) |
| 37 | \( 1 + 5.01T + 37T^{2} \) |
| 41 | \( 1 - 2.83T + 41T^{2} \) |
| 43 | \( 1 - 1.12T + 43T^{2} \) |
| 47 | \( 1 - 3.36T + 47T^{2} \) |
| 53 | \( 1 - 3.21T + 53T^{2} \) |
| 59 | \( 1 - 11.6T + 59T^{2} \) |
| 61 | \( 1 + 10.7T + 61T^{2} \) |
| 67 | \( 1 - 0.456T + 67T^{2} \) |
| 71 | \( 1 + 0.755T + 71T^{2} \) |
| 73 | \( 1 + 2.76T + 73T^{2} \) |
| 79 | \( 1 - 12.9T + 79T^{2} \) |
| 83 | \( 1 + 5.20T + 83T^{2} \) |
| 89 | \( 1 + 7.13T + 89T^{2} \) |
| 97 | \( 1 - 9.29T + 97T^{2} \) |
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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
−7.976288842653202021415525927247, −7.46619067410226765575987730415, −6.64747666069475587947643912271, −5.63879230012976102969926772519, −4.98355844909025276269660094479, −4.34510605105925246236161170614, −3.50627502669514081606679201628, −2.54283855602542589749492324482, −1.50106378325660313823356907781, −0.936255615934757949010764790315,
0.936255615934757949010764790315, 1.50106378325660313823356907781, 2.54283855602542589749492324482, 3.50627502669514081606679201628, 4.34510605105925246236161170614, 4.98355844909025276269660094479, 5.63879230012976102969926772519, 6.64747666069475587947643912271, 7.46619067410226765575987730415, 7.976288842653202021415525927247
