L(s) = 1 | + 2·2-s + 3-s + 2·4-s + 5-s + 2·6-s − 2·7-s − 2·9-s + 2·10-s + 11-s + 2·12-s − 4·13-s − 4·14-s + 15-s − 4·16-s − 2·17-s − 4·18-s + 2·20-s − 2·21-s + 2·22-s − 23-s − 4·25-s − 8·26-s − 5·27-s − 4·28-s + 2·30-s − 7·31-s − 8·32-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 0.577·3-s + 4-s + 0.447·5-s + 0.816·6-s − 0.755·7-s − 2/3·9-s + 0.632·10-s + 0.301·11-s + 0.577·12-s − 1.10·13-s − 1.06·14-s + 0.258·15-s − 16-s − 0.485·17-s − 0.942·18-s + 0.447·20-s − 0.436·21-s + 0.426·22-s − 0.208·23-s − 4/5·25-s − 1.56·26-s − 0.962·27-s − 0.755·28-s + 0.365·30-s − 1.25·31-s − 1.41·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3971 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3971 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 - T \) |
| 19 | \( 1 \) |
good | 2 | \( 1 - p T + p T^{2} \) |
| 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−7.991382198601661869330644512838, −7.08610298939652805070973785571, −6.44705964567155032403967809781, −5.65921461221479271613429705311, −5.19096686691774806327844897047, −4.09468272193076097429923581775, −3.55069629742260821889174578377, −2.62758010025505468758993615534, −2.12638900923174362608185387010, 0,
2.12638900923174362608185387010, 2.62758010025505468758993615534, 3.55069629742260821889174578377, 4.09468272193076097429923581775, 5.19096686691774806327844897047, 5.65921461221479271613429705311, 6.44705964567155032403967809781, 7.08610298939652805070973785571, 7.991382198601661869330644512838