| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s + 2·7-s + 8-s + 9-s − 10-s − 12-s + 2·14-s + 15-s + 16-s − 17-s + 18-s − 20-s − 2·21-s − 4·23-s − 24-s + 25-s − 27-s + 2·28-s − 2·29-s + 30-s + 4·31-s + 32-s − 34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.755·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.288·12-s + 0.534·14-s + 0.258·15-s + 1/4·16-s − 0.242·17-s + 0.235·18-s − 0.223·20-s − 0.436·21-s − 0.834·23-s − 0.204·24-s + 1/5·25-s − 0.192·27-s + 0.377·28-s − 0.371·29-s + 0.182·30-s + 0.718·31-s + 0.176·32-s − 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 61710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−14.63031867177991, −13.99510451523658, −13.60267591815603, −12.84178480172067, −12.59870929804820, −11.91773049864323, −11.56442050297616, −11.13010922362602, −10.74155420942737, −9.937568103627447, −9.703975180773519, −8.530650339947738, −8.448220269938256, −7.616662022036926, −7.213329130099336, −6.623109373269739, −5.988351421343137, −5.501975667331152, −4.940527565110534, −4.315423558340183, −4.013402235801823, −3.199514709761634, −2.442283590078994, −1.754309005162276, −1.010798257633693, 0,
1.010798257633693, 1.754309005162276, 2.442283590078994, 3.199514709761634, 4.013402235801823, 4.315423558340183, 4.940527565110534, 5.501975667331152, 5.988351421343137, 6.623109373269739, 7.213329130099336, 7.616662022036926, 8.448220269938256, 8.530650339947738, 9.703975180773519, 9.937568103627447, 10.74155420942737, 11.13010922362602, 11.56442050297616, 11.91773049864323, 12.59870929804820, 12.84178480172067, 13.60267591815603, 13.99510451523658, 14.63031867177991
