| L(s) = 1 | − 2-s + 4-s − 4·5-s − 8-s + 4·10-s + 11-s + 6·13-s + 16-s − 4·17-s + 2·19-s − 4·20-s − 22-s + 8·23-s + 11·25-s − 6·26-s + 6·29-s − 6·31-s − 32-s + 4·34-s − 6·37-s − 2·38-s + 4·40-s + 12·41-s + 4·43-s + 44-s − 8·46-s + 6·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.78·5-s − 0.353·8-s + 1.26·10-s + 0.301·11-s + 1.66·13-s + 1/4·16-s − 0.970·17-s + 0.458·19-s − 0.894·20-s − 0.213·22-s + 1.66·23-s + 11/5·25-s − 1.17·26-s + 1.11·29-s − 1.07·31-s − 0.176·32-s + 0.685·34-s − 0.986·37-s − 0.324·38-s + 0.632·40-s + 1.87·41-s + 0.609·43-s + 0.150·44-s − 1.17·46-s + 0.875·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.073848941\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.073848941\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−7.63555539558063109991704699013, −7.19600541996782844068565480276, −6.59545968662394864794844007037, −5.78492019542310094876621133267, −4.74280139968093732696458509052, −4.06271186650227623738805370448, −3.44347171901910994813953620793, −2.73256133422613809320045014753, −1.33009229045687298156582080211, −0.61967504270051838638096106576,
0.61967504270051838638096106576, 1.33009229045687298156582080211, 2.73256133422613809320045014753, 3.44347171901910994813953620793, 4.06271186650227623738805370448, 4.74280139968093732696458509052, 5.78492019542310094876621133267, 6.59545968662394864794844007037, 7.19600541996782844068565480276, 7.63555539558063109991704699013
