L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 8-s + 9-s + 10-s − 12-s − 13-s + 15-s + 16-s − 2·17-s − 18-s + 19-s − 20-s − 4·23-s + 24-s + 25-s + 26-s − 27-s + 2·29-s − 30-s − 32-s + 2·34-s + 36-s + 6·37-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.288·12-s − 0.277·13-s + 0.258·15-s + 1/4·16-s − 0.485·17-s − 0.235·18-s + 0.229·19-s − 0.223·20-s − 0.834·23-s + 0.204·24-s + 1/5·25-s + 0.196·26-s − 0.192·27-s + 0.371·29-s − 0.182·30-s − 0.176·32-s + 0.342·34-s + 1/6·36-s + 0.986·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363090 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363090 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| 19 | \( 1 - T \) |
good | 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−12.53413311662608, −12.16158659666325, −11.72122798906829, −11.41361022795198, −10.95988029008848, −10.45893751818832, −10.03079360671180, −9.613873996581512, −9.224367012087254, −8.533796880614851, −8.116009866919805, −7.861535020262428, −7.203429627757368, −6.727722488959738, −6.417135392530075, −5.877361578449533, −5.274483285947743, −4.698092396871408, −4.421170545109236, −3.531830416941156, −3.263968293055723, −2.465306145163618, −1.874284305413646, −1.353738273685788, −0.5487330267595959, 0,
0.5487330267595959, 1.353738273685788, 1.874284305413646, 2.465306145163618, 3.263968293055723, 3.531830416941156, 4.421170545109236, 4.698092396871408, 5.274483285947743, 5.877361578449533, 6.417135392530075, 6.727722488959738, 7.203429627757368, 7.861535020262428, 8.116009866919805, 8.533796880614851, 9.224367012087254, 9.613873996581512, 10.03079360671180, 10.45893751818832, 10.95988029008848, 11.41361022795198, 11.72122798906829, 12.16158659666325, 12.53413311662608