L(s) = 1 | + 1.41i·5-s + i·7-s + 9-s + 1.41·11-s + 1.41i·13-s − 1.00·25-s − 2i·31-s − 1.41·35-s − 1.41·43-s + 1.41i·45-s − 2i·47-s − 49-s + 2.00i·55-s − 1.41i·61-s + i·63-s + ⋯ |
L(s) = 1 | + 1.41i·5-s + i·7-s + 9-s + 1.41·11-s + 1.41i·13-s − 1.00·25-s − 2i·31-s − 1.41·35-s − 1.41·43-s + 1.41i·45-s − 2i·47-s − 49-s + 2.00i·55-s − 1.41i·61-s + i·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.505782148\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.505782148\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 - iT \) |
good | 3 | \( 1 - T^{2} \) |
| 5 | \( 1 - 1.41iT - T^{2} \) |
| 11 | \( 1 - 1.41T + T^{2} \) |
| 13 | \( 1 - 1.41iT - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + 2iT - T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + 1.41T + T^{2} \) |
| 47 | \( 1 + 2iT - T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + 1.41iT - T^{2} \) |
| 67 | \( 1 - 1.41T + T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.105258970196171148313678916503, −8.154480411533543398088357932795, −7.21477000880549414506215253583, −6.50591656523108467006499353494, −6.42095431891894210900164211972, −5.15802516343223768273212559177, −4.07652877546709627477569665315, −3.59117934576752853411265059366, −2.34670263617491329562798163291, −1.73748507093981321275491737274,
1.04758633771778830653930507040, 1.43783836857287208080680197779, 3.23574374349493958232952753058, 4.04727584697386713694811014306, 4.67403151682448148823671157555, 5.34530639815984334354366400641, 6.43146408526098419550087357152, 7.08615100044309774153737209664, 7.889196125691832539438283410405, 8.542161051754276076324776931476