L(s) = 1 | + 2-s + 4-s − 5-s − 7-s + 8-s − 10-s − 3·11-s − 13-s − 14-s + 16-s − 5·17-s + 19-s − 20-s − 3·22-s − 3·23-s − 4·25-s − 26-s − 28-s − 5·29-s + 4·31-s + 32-s − 5·34-s + 35-s − 5·37-s + 38-s − 40-s + 8·41-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 − 0.904·11-s − 0.277·13-s − 0.267·14-s + 1/4·16-s − 1.21·17-s + 0.229·19-s − 0.223·20-s − 0.639·22-s − 0.625·23-s − 4/5·25-s − 0.196·26-s − 0.188·28-s − 0.928·29-s + 0.718·31-s + 0.176·32-s − 0.857·34-s + 0.169·35-s − 0.821·37-s + 0.162·38-s − 0.158·40-s + 1.24·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 + T \) |
good | 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 15 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.960650688894051452457795645047, −7.983338956369825188614913231789, −7.37916278392494576475595880121, −6.46749178501716904289122376294, −5.67367236682380813987886583185, −4.74733794167405663778435697848, −3.96769573665723182148627582648, −2.97754543141818136913211308555, −2.00705820570952324674068994753, 0,
2.00705820570952324674068994753, 2.97754543141818136913211308555, 3.96769573665723182148627582648, 4.74733794167405663778435697848, 5.67367236682380813987886583185, 6.46749178501716904289122376294, 7.37916278392494576475595880121, 7.983338956369825188614913231789, 8.960650688894051452457795645047