L(s) = 1 | + 2-s + 2·3-s + 4-s + 2·6-s + 4·7-s + 8-s + 9-s − 6·11-s + 2·12-s − 2·13-s + 4·14-s + 16-s + 17-s + 18-s + 4·19-s + 8·21-s − 6·22-s + 2·24-s − 5·25-s − 2·26-s − 4·27-s + 4·28-s − 4·31-s + 32-s − 12·33-s + 34-s + 36-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.816·6-s + 1.51·7-s + 0.353·8-s + 1/3·9-s − 1.80·11-s + 0.577·12-s − 0.554·13-s + 1.06·14-s + 1/4·16-s + 0.242·17-s + 0.235·18-s + 0.917·19-s + 1.74·21-s − 1.27·22-s + 0.408·24-s − 25-s − 0.392·26-s − 0.769·27-s + 0.755·28-s − 0.718·31-s + 0.176·32-s − 2.08·33-s + 0.171·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57154 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57154 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 41 | \( 1 \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−14.62185559294523, −14.06211778736626, −13.63295205751870, −13.45411002088285, −12.61482313036287, −12.23693262933721, −11.54784819264918, −11.16773467922447, −10.53474564454704, −10.12100730607322, −9.395727575003135, −8.861483320753064, −8.087352211831896, −7.901812406642175, −7.498958101598048, −7.013399999192559, −5.768283901901817, −5.486044816446002, −5.042830943992067, −4.345793334911463, −3.777493064051272, −2.972360943340543, −2.557318483107616, −2.015136684079117, −1.335556577308960, 0,
1.335556577308960, 2.015136684079117, 2.557318483107616, 2.972360943340543, 3.777493064051272, 4.345793334911463, 5.042830943992067, 5.486044816446002, 5.768283901901817, 7.013399999192559, 7.498958101598048, 7.901812406642175, 8.087352211831896, 8.861483320753064, 9.395727575003135, 10.12100730607322, 10.53474564454704, 11.16773467922447, 11.54784819264918, 12.23693262933721, 12.61482313036287, 13.45411002088285, 13.63295205751870, 14.06211778736626, 14.62185559294523