L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 8-s + 9-s − 4·11-s + 12-s − 6·13-s + 16-s − 6·17-s + 18-s + 8·19-s − 4·22-s + 24-s − 6·26-s + 27-s − 6·29-s + 4·31-s + 32-s − 4·33-s − 6·34-s + 36-s − 37-s + 8·38-s − 6·39-s − 6·41-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.353·8-s + 1/3·9-s − 1.20·11-s + 0.288·12-s − 1.66·13-s + 1/4·16-s − 1.45·17-s + 0.235·18-s + 1.83·19-s − 0.852·22-s + 0.204·24-s − 1.17·26-s + 0.192·27-s − 1.11·29-s + 0.718·31-s + 0.176·32-s − 0.696·33-s − 1.02·34-s + 1/6·36-s − 0.164·37-s + 1.29·38-s − 0.960·39-s − 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 37 | \( 1 + T \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−7.61601947318365482592160501957, −7.22666994333741871570583012402, −6.40464464067345167290601939623, −5.28072637503090546048953297022, −4.99252953685096340742249822443, −4.15515931791529816551942149312, −3.05864596219030005698298261579, −2.63609737900778525221072071588, −1.72612495106527709578247299004, 0,
1.72612495106527709578247299004, 2.63609737900778525221072071588, 3.05864596219030005698298261579, 4.15515931791529816551942149312, 4.99252953685096340742249822443, 5.28072637503090546048953297022, 6.40464464067345167290601939623, 7.22666994333741871570583012402, 7.61601947318365482592160501957