| L(s) = 1 | − 2-s + 4-s − 5-s + 7-s − 8-s + 10-s − 11-s + 13-s − 14-s + 16-s − 17-s − 7·19-s − 20-s + 22-s + 3·23-s + 25-s − 26-s + 28-s + 8·29-s − 5·31-s − 32-s + 34-s − 35-s + 5·37-s + 7·38-s + 40-s + 10·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.377·7-s − 0.353·8-s + 0.316·10-s − 0.301·11-s + 0.277·13-s − 0.267·14-s + 1/4·16-s − 0.242·17-s − 1.60·19-s − 0.223·20-s + 0.213·22-s + 0.625·23-s + 1/5·25-s − 0.196·26-s + 0.188·28-s + 1.48·29-s − 0.898·31-s − 0.176·32-s + 0.171·34-s − 0.169·35-s + 0.821·37-s + 1.13·38-s + 0.158·40-s + 1.56·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16830 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16830 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 9 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
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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
−16.16038522093195, −15.76862360539629, −15.07402256088265, −14.68970350306790, −14.13402319789308, −13.28215702919422, −12.69784724483732, −12.37515521334948, −11.40223963043811, −11.15236810435491, −10.59431586072650, −10.06367182694671, −9.223215775700059, −8.727982832654222, −8.281107682624322, −7.594560109032239, −7.156702813516730, −6.206132558672556, −5.993665477554194, −4.668096565152519, −4.528714814286349, −3.439439748535999, −2.704625120985872, −1.936800877464726, −1.008805138413670, 0,
1.008805138413670, 1.936800877464726, 2.704625120985872, 3.439439748535999, 4.528714814286349, 4.668096565152519, 5.993665477554194, 6.206132558672556, 7.156702813516730, 7.594560109032239, 8.281107682624322, 8.727982832654222, 9.223215775700059, 10.06367182694671, 10.59431586072650, 11.15236810435491, 11.40223963043811, 12.37515521334948, 12.69784724483732, 13.28215702919422, 14.13402319789308, 14.68970350306790, 15.07402256088265, 15.76862360539629, 16.16038522093195
