L(s) = 1 | − 2-s − 3-s + 4-s + 4·5-s + 6-s + 7-s − 8-s + 9-s − 4·10-s − 12-s − 2·13-s − 14-s − 4·15-s + 16-s + 3·17-s − 18-s + 4·20-s − 21-s − 23-s + 24-s + 11·25-s + 2·26-s − 27-s + 28-s + 5·29-s + 4·30-s + 5·31-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 1.78·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 1.26·10-s − 0.288·12-s − 0.554·13-s − 0.267·14-s − 1.03·15-s + 1/4·16-s + 0.727·17-s − 0.235·18-s + 0.894·20-s − 0.218·21-s − 0.208·23-s + 0.204·24-s + 11/5·25-s + 0.392·26-s − 0.192·27-s + 0.188·28-s + 0.928·29-s + 0.730·30-s + 0.898·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 348726 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 348726 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 \) |
| 23 | \( 1 + T \) |
good | 5 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 3 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−12.62755243281270, −12.26513029254791, −11.90788793245374, −11.35967455060441, −10.71137886487319, −10.42332324021526, −10.02688706479385, −9.794094197332491, −9.189418970078367, −8.769860874305145, −8.334928662024868, −7.673694150506398, −7.200723020472302, −6.695864175138890, −6.328207575213636, −5.702208081770374, −5.494076089262330, −4.971704588690134, −4.418432922161381, −3.693591653204695, −2.779727928510252, −2.541984397854659, −1.927503764950253, −1.238792561684424, −1.011637512996279, 0,
1.011637512996279, 1.238792561684424, 1.927503764950253, 2.541984397854659, 2.779727928510252, 3.693591653204695, 4.418432922161381, 4.971704588690134, 5.494076089262330, 5.702208081770374, 6.328207575213636, 6.695864175138890, 7.200723020472302, 7.673694150506398, 8.334928662024868, 8.769860874305145, 9.189418970078367, 9.794094197332491, 10.02688706479385, 10.42332324021526, 10.71137886487319, 11.35967455060441, 11.90788793245374, 12.26513029254791, 12.62755243281270