L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 7-s + 8-s − 2·9-s + 12-s − 5·13-s + 14-s + 16-s − 3·17-s − 2·18-s − 19-s + 21-s + 3·23-s + 24-s − 5·25-s − 5·26-s − 5·27-s + 28-s − 9·29-s − 4·31-s + 32-s − 3·34-s − 2·36-s + 2·37-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.377·7-s + 0.353·8-s − 2/3·9-s + 0.288·12-s − 1.38·13-s + 0.267·14-s + 1/4·16-s − 0.727·17-s − 0.471·18-s − 0.229·19-s + 0.218·21-s + 0.625·23-s + 0.204·24-s − 25-s − 0.980·26-s − 0.962·27-s + 0.188·28-s − 1.67·29-s − 0.718·31-s + 0.176·32-s − 0.514·34-s − 1/3·36-s + 0.328·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−7.88773793776252745940024787045, −7.25858039362668764589828470989, −6.52390816581107389509679557007, −5.50569298727080846748906445848, −5.08583557668970609096298687159, −4.11581948005602868281874533970, −3.39344369775766583219012744185, −2.42504146949326223577793697879, −1.89915892713560405275098298657, 0,
1.89915892713560405275098298657, 2.42504146949326223577793697879, 3.39344369775766583219012744185, 4.11581948005602868281874533970, 5.08583557668970609096298687159, 5.50569298727080846748906445848, 6.52390816581107389509679557007, 7.25858039362668764589828470989, 7.88773793776252745940024787045