| L(s) = 1 | − 3-s − 5-s + 2·7-s + 9-s + 13-s + 15-s − 17-s − 2·21-s + 25-s − 27-s − 2·29-s + 2·31-s − 2·35-s + 8·37-s − 39-s + 2·41-s − 4·43-s − 45-s − 6·47-s − 3·49-s + 51-s + 2·53-s + 2·59-s − 10·61-s + 2·63-s − 65-s + 4·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.755·7-s + 1/3·9-s + 0.277·13-s + 0.258·15-s − 0.242·17-s − 0.436·21-s + 1/5·25-s − 0.192·27-s − 0.371·29-s + 0.359·31-s − 0.338·35-s + 1.31·37-s − 0.160·39-s + 0.312·41-s − 0.609·43-s − 0.149·45-s − 0.875·47-s − 3/7·49-s + 0.140·51-s + 0.274·53-s + 0.260·59-s − 1.28·61-s + 0.251·63-s − 0.124·65-s + 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−15.51100339625109, −15.09990195205043, −14.53063033129549, −14.06762571680509, −13.18086452869658, −13.03763832254445, −12.17147863615974, −11.75344889555165, −11.20466022379755, −10.94373193158637, −10.18278216263964, −9.655618303852058, −8.923694830629873, −8.355545016926837, −7.774984610448705, −7.329508271816564, −6.513272197381213, −6.079733882347197, −5.301574124625093, −4.727048611884678, −4.236353067185309, −3.490495687505243, −2.656674469415320, −1.750775518372726, −1.026283271148462, 0,
1.026283271148462, 1.750775518372726, 2.656674469415320, 3.490495687505243, 4.236353067185309, 4.727048611884678, 5.301574124625093, 6.079733882347197, 6.513272197381213, 7.329508271816564, 7.774984610448705, 8.355545016926837, 8.923694830629873, 9.655618303852058, 10.18278216263964, 10.94373193158637, 11.20466022379755, 11.75344889555165, 12.17147863615974, 13.03763832254445, 13.18086452869658, 14.06762571680509, 14.53063033129549, 15.09990195205043, 15.51100339625109
