| L(s) = 1 | − 2-s + 4-s − 7-s − 8-s + 2·13-s + 14-s + 16-s − 2·17-s − 4·19-s − 4·23-s − 2·26-s − 28-s − 6·29-s − 32-s + 2·34-s − 2·37-s + 4·38-s + 6·41-s + 12·43-s + 4·46-s + 8·47-s + 49-s + 2·52-s − 6·53-s + 56-s + 6·58-s − 8·59-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s + 0.554·13-s + 0.267·14-s + 1/4·16-s − 0.485·17-s − 0.917·19-s − 0.834·23-s − 0.392·26-s − 0.188·28-s − 1.11·29-s − 0.176·32-s + 0.342·34-s − 0.328·37-s + 0.648·38-s + 0.937·41-s + 1.82·43-s + 0.589·46-s + 1.16·47-s + 1/7·49-s + 0.277·52-s − 0.824·53-s + 0.133·56-s + 0.787·58-s − 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 381150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 381150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−12.49707511519547, −12.44607083913454, −11.64754339183697, −11.20331427429896, −10.88997239270771, −10.45826790824786, −9.996719320220694, −9.443694156082693, −9.033459648364299, −8.783894582243218, −8.086116086627213, −7.819569286135028, −7.160199879989786, −6.856754197905864, −6.229565849868327, −5.738481885885520, −5.612601287897895, −4.553131983670856, −4.129398641385528, −3.799853005910118, −2.961461288410166, −2.547568089497935, −1.934342818871115, −1.418649571687250, −0.6162790447090462, 0,
0.6162790447090462, 1.418649571687250, 1.934342818871115, 2.547568089497935, 2.961461288410166, 3.799853005910118, 4.129398641385528, 4.553131983670856, 5.612601287897895, 5.738481885885520, 6.229565849868327, 6.856754197905864, 7.160199879989786, 7.819569286135028, 8.086116086627213, 8.783894582243218, 9.033459648364299, 9.443694156082693, 9.996719320220694, 10.45826790824786, 10.88997239270771, 11.20331427429896, 11.64754339183697, 12.44607083913454, 12.49707511519547
