| L(s) = 1 | + 0.852·2-s − 3-s − 1.27·4-s + 2.61·5-s − 0.852·6-s − 1.65·7-s − 2.79·8-s + 9-s + 2.22·10-s − 11-s + 1.27·12-s + 2.92·13-s − 1.41·14-s − 2.61·15-s + 0.164·16-s + 1.17·17-s + 0.852·18-s − 6.88·19-s − 3.32·20-s + 1.65·21-s − 0.852·22-s − 1.74·23-s + 2.79·24-s + 1.82·25-s + 2.49·26-s − 27-s + 2.10·28-s + ⋯ |
| L(s) = 1 | + 0.603·2-s − 0.577·3-s − 0.636·4-s + 1.16·5-s − 0.348·6-s − 0.626·7-s − 0.986·8-s + 0.333·9-s + 0.704·10-s − 0.301·11-s + 0.367·12-s + 0.811·13-s − 0.377·14-s − 0.674·15-s + 0.0411·16-s + 0.285·17-s + 0.201·18-s − 1.57·19-s − 0.743·20-s + 0.361·21-s − 0.181·22-s − 0.364·23-s + 0.569·24-s + 0.365·25-s + 0.489·26-s − 0.192·27-s + 0.398·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 - 0.852T + 2T^{2} \) |
| 5 | \( 1 - 2.61T + 5T^{2} \) |
| 7 | \( 1 + 1.65T + 7T^{2} \) |
| 13 | \( 1 - 2.92T + 13T^{2} \) |
| 17 | \( 1 - 1.17T + 17T^{2} \) |
| 19 | \( 1 + 6.88T + 19T^{2} \) |
| 23 | \( 1 + 1.74T + 23T^{2} \) |
| 29 | \( 1 - 3.91T + 29T^{2} \) |
| 31 | \( 1 + 5.07T + 31T^{2} \) |
| 37 | \( 1 - 6.35T + 37T^{2} \) |
| 41 | \( 1 + 11.5T + 41T^{2} \) |
| 43 | \( 1 - 1.20T + 43T^{2} \) |
| 47 | \( 1 + 11.3T + 47T^{2} \) |
| 53 | \( 1 - 3.35T + 53T^{2} \) |
| 59 | \( 1 + 5.28T + 59T^{2} \) |
| 67 | \( 1 - 4.96T + 67T^{2} \) |
| 71 | \( 1 + 8.80T + 71T^{2} \) |
| 73 | \( 1 + 2.70T + 73T^{2} \) |
| 79 | \( 1 + 11.4T + 79T^{2} \) |
| 83 | \( 1 - 6.15T + 83T^{2} \) |
| 89 | \( 1 + 14.3T + 89T^{2} \) |
| 97 | \( 1 + 10.1T + 97T^{2} \) |
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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
−8.828893066437444557037640751712, −8.160999710738067710995618404481, −6.71085348944682282994738797763, −6.17526626862832706782698983356, −5.64355410673635158927722727964, −4.79581781712804995753305572031, −3.92240648486491013105417472878, −2.92185280490503461281589994336, −1.64605151460735592901237346026, 0,
1.64605151460735592901237346026, 2.92185280490503461281589994336, 3.92240648486491013105417472878, 4.79581781712804995753305572031, 5.64355410673635158927722727964, 6.17526626862832706782698983356, 6.71085348944682282994738797763, 8.160999710738067710995618404481, 8.828893066437444557037640751712
