| L(s) = 1 | − 2-s + 4-s − 5-s − 2·7-s − 8-s + 10-s + 4·11-s + 13-s + 2·14-s + 16-s + 2·19-s − 20-s − 4·22-s − 8·23-s + 25-s − 26-s − 2·28-s − 2·29-s − 9·31-s − 32-s + 2·35-s − 6·37-s − 2·38-s + 40-s − 6·41-s + 4·43-s + 4·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.755·7-s − 0.353·8-s + 0.316·10-s + 1.20·11-s + 0.277·13-s + 0.534·14-s + 1/4·16-s + 0.458·19-s − 0.223·20-s − 0.852·22-s − 1.66·23-s + 1/5·25-s − 0.196·26-s − 0.377·28-s − 0.371·29-s − 1.61·31-s − 0.176·32-s + 0.338·35-s − 0.986·37-s − 0.324·38-s + 0.158·40-s − 0.937·41-s + 0.609·43-s + 0.603·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7854750148\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7854750148\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 9 T + p T^{2} \) |
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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
−12.34656076142318, −12.06268402930688, −11.77157368164671, −11.12598219265047, −10.78315621396569, −10.23589243265547, −9.791944076134050, −9.326010724490858, −9.000450860943125, −8.558853135357003, −7.984173734101548, −7.511546967277384, −7.061187507293026, −6.648475564816154, −6.118711460502798, −5.729973358953938, −5.145289184255771, −4.345404101532140, −3.722262011405161, −3.622876939416442, −2.979763093691466, −2.072900695148416, −1.772349119688979, −0.9958474465718613, −0.2934165226744504,
0.2934165226744504, 0.9958474465718613, 1.772349119688979, 2.072900695148416, 2.979763093691466, 3.622876939416442, 3.722262011405161, 4.345404101532140, 5.145289184255771, 5.729973358953938, 6.118711460502798, 6.648475564816154, 7.061187507293026, 7.511546967277384, 7.984173734101548, 8.558853135357003, 9.000450860943125, 9.326010724490858, 9.791944076134050, 10.23589243265547, 10.78315621396569, 11.12598219265047, 11.77157368164671, 12.06268402930688, 12.34656076142318
