| L(s) = 1 | + 3·3-s + 5-s + 7-s + 6·9-s + 5·11-s + 3·15-s − 17-s − 6·19-s + 3·21-s + 6·23-s + 25-s + 9·27-s − 9·29-s + 4·31-s + 15·33-s + 35-s − 2·37-s + 4·41-s + 10·43-s + 6·45-s + 47-s + 49-s − 3·51-s + 4·53-s + 5·55-s − 18·57-s + 8·59-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 0.447·5-s + 0.377·7-s + 2·9-s + 1.50·11-s + 0.774·15-s − 0.242·17-s − 1.37·19-s + 0.654·21-s + 1.25·23-s + 1/5·25-s + 1.73·27-s − 1.67·29-s + 0.718·31-s + 2.61·33-s + 0.169·35-s − 0.328·37-s + 0.624·41-s + 1.52·43-s + 0.894·45-s + 0.145·47-s + 1/7·49-s − 0.420·51-s + 0.549·53-s + 0.674·55-s − 2.38·57-s + 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23660 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23660 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.471319556\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.471319556\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 13 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−15.10829497605217, −14.87304634328965, −14.35960861551818, −14.02987966753173, −13.25667179579135, −13.05598628057021, −12.37007366475754, −11.66738779502338, −10.95781059586753, −10.46033288481374, −9.699152723243973, −9.059658457515939, −8.963904568179028, −8.468544846038525, −7.495623273033405, −7.315476413584121, −6.447306065896133, −5.952064375288537, −4.864168929855440, −4.222147299526863, −3.791527681721566, −3.027440649229496, −2.260824470015868, −1.783605921416264, −0.9736410719570755,
0.9736410719570755, 1.783605921416264, 2.260824470015868, 3.027440649229496, 3.791527681721566, 4.222147299526863, 4.864168929855440, 5.952064375288537, 6.447306065896133, 7.315476413584121, 7.495623273033405, 8.468544846038525, 8.963904568179028, 9.059658457515939, 9.699152723243973, 10.46033288481374, 10.95781059586753, 11.66738779502338, 12.37007366475754, 13.05598628057021, 13.25667179579135, 14.02987966753173, 14.35960861551818, 14.87304634328965, 15.10829497605217
