L(s) = 1 | − 2-s + 4-s − 5-s + 7-s − 8-s − 3·9-s + 10-s + 2·11-s − 4·13-s − 14-s + 16-s + 6·17-s + 3·18-s − 4·19-s − 20-s − 2·22-s + 25-s + 4·26-s + 28-s + 6·29-s − 10·31-s − 32-s − 6·34-s − 35-s − 3·36-s − 6·37-s + 4·38-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.377·7-s − 0.353·8-s − 9-s + 0.316·10-s + 0.603·11-s − 1.10·13-s − 0.267·14-s + 1/4·16-s + 1.45·17-s + 0.707·18-s − 0.917·19-s − 0.223·20-s − 0.426·22-s + 1/5·25-s + 0.784·26-s + 0.188·28-s + 1.11·29-s − 1.79·31-s − 0.176·32-s − 1.02·34-s − 0.169·35-s − 1/2·36-s − 0.986·37-s + 0.648·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 \) |
good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−15.02173060265151, −14.58515219290810, −14.33296801647390, −13.84248700031516, −12.65887624585805, −12.48329028902218, −11.94322338118959, −11.42841059296908, −10.88573406352604, −10.45300438857175, −9.731832038145978, −9.269672877206543, −8.690710343146941, −8.152170290518775, −7.743100018000604, −7.096738841366325, −6.607248288201106, −5.746378985856441, −5.372363080123369, −4.600515857678347, −3.819273993313979, −3.184112116869335, −2.488128823215843, −1.775662376510138, −0.8459942035292666, 0,
0.8459942035292666, 1.775662376510138, 2.488128823215843, 3.184112116869335, 3.819273993313979, 4.600515857678347, 5.372363080123369, 5.746378985856441, 6.607248288201106, 7.096738841366325, 7.743100018000604, 8.152170290518775, 8.690710343146941, 9.269672877206543, 9.731832038145978, 10.45300438857175, 10.88573406352604, 11.42841059296908, 11.94322338118959, 12.48329028902218, 12.65887624585805, 13.84248700031516, 14.33296801647390, 14.58515219290810, 15.02173060265151