| L(s) = 1 | − 2·2-s + 2·4-s − 5-s + 7-s + 2·10-s − 6·13-s − 2·14-s − 4·16-s + 7·17-s − 8·19-s − 2·20-s − 6·23-s − 4·25-s + 12·26-s + 2·28-s − 4·29-s + 2·31-s + 8·32-s − 14·34-s − 35-s − 6·37-s + 16·38-s + 2·41-s − 43-s + 12·46-s − 13·47-s + 49-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s − 0.447·5-s + 0.377·7-s + 0.632·10-s − 1.66·13-s − 0.534·14-s − 16-s + 1.69·17-s − 1.83·19-s − 0.447·20-s − 1.25·23-s − 4/5·25-s + 2.35·26-s + 0.377·28-s − 0.742·29-s + 0.359·31-s + 1.41·32-s − 2.40·34-s − 0.169·35-s − 0.986·37-s + 2.59·38-s + 0.312·41-s − 0.152·43-s + 1.76·46-s − 1.89·47-s + 1/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3231837849\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3231837849\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 13 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 5 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−7.976943931772590121480041500586, −7.54822384140609894076207099120, −6.80665812374789933283025781741, −5.96600466270722379176114281353, −5.01323940502938286833247864544, −4.35723196397667319235527539918, −3.44566668797560661682609106999, −2.23643956111815514723991654657, −1.71394393522637452268109773975, −0.34437697388338477839333244307,
0.34437697388338477839333244307, 1.71394393522637452268109773975, 2.23643956111815514723991654657, 3.44566668797560661682609106999, 4.35723196397667319235527539918, 5.01323940502938286833247864544, 5.96600466270722379176114281353, 6.80665812374789933283025781741, 7.54822384140609894076207099120, 7.976943931772590121480041500586
