L(s) = 1 | + 2·2-s − 7·3-s + 4·4-s + 5·5-s − 14·6-s − 2·7-s + 8·8-s + 22·9-s + 10·10-s − 37·11-s − 28·12-s + 27·13-s − 4·14-s − 35·15-s + 16·16-s − 24·17-s + 44·18-s + 88·19-s + 20·20-s + 14·21-s − 74·22-s − 28·23-s − 56·24-s − 100·25-s + 54·26-s + 35·27-s − 8·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.34·3-s + 1/2·4-s + 0.447·5-s − 0.952·6-s − 0.107·7-s + 0.353·8-s + 0.814·9-s + 0.316·10-s − 1.01·11-s − 0.673·12-s + 0.576·13-s − 0.0763·14-s − 0.602·15-s + 1/4·16-s − 0.342·17-s + 0.576·18-s + 1.06·19-s + 0.223·20-s + 0.145·21-s − 0.717·22-s − 0.253·23-s − 0.476·24-s − 4/5·25-s + 0.407·26-s + 0.249·27-s − 0.0539·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1682 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1682 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - p T \) |
| 29 | \( 1 \) |
good | 3 | \( 1 + 7 T + p^{3} T^{2} \) |
| 5 | \( 1 - p T + p^{3} T^{2} \) |
| 7 | \( 1 + 2 T + p^{3} T^{2} \) |
| 11 | \( 1 + 37 T + p^{3} T^{2} \) |
| 13 | \( 1 - 27 T + p^{3} T^{2} \) |
| 17 | \( 1 + 24 T + p^{3} T^{2} \) |
| 19 | \( 1 - 88 T + p^{3} T^{2} \) |
| 23 | \( 1 + 28 T + p^{3} T^{2} \) |
| 31 | \( 1 - 143 T + p^{3} T^{2} \) |
| 37 | \( 1 - 360 T + p^{3} T^{2} \) |
| 41 | \( 1 + 386 T + p^{3} T^{2} \) |
| 43 | \( 1 + 381 T + p^{3} T^{2} \) |
| 47 | \( 1 - 103 T + p^{3} T^{2} \) |
| 53 | \( 1 + 431 T + p^{3} T^{2} \) |
| 59 | \( 1 - 288 T + p^{3} T^{2} \) |
| 61 | \( 1 - 840 T + p^{3} T^{2} \) |
| 67 | \( 1 + 180 T + p^{3} T^{2} \) |
| 71 | \( 1 - 706 T + p^{3} T^{2} \) |
| 73 | \( 1 + 716 T + p^{3} T^{2} \) |
| 79 | \( 1 + 931 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1188 T + p^{3} T^{2} \) |
| 89 | \( 1 - 642 T + p^{3} T^{2} \) |
| 97 | \( 1 + 486 T + p^{3} T^{2} \) |
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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
−8.441617123513745835688644171747, −7.59965450257279681288745896696, −6.55439064006180327721043259127, −6.08857962856197489945599053109, −5.27703245772853535763835974397, −4.81400806237131701710144433923, −3.60038134971572399454830269143, −2.50844294831584910969985138310, −1.25004343657418131865356580716, 0,
1.25004343657418131865356580716, 2.50844294831584910969985138310, 3.60038134971572399454830269143, 4.81400806237131701710144433923, 5.27703245772853535763835974397, 6.08857962856197489945599053109, 6.55439064006180327721043259127, 7.59965450257279681288745896696, 8.441617123513745835688644171747