| L(s) = 1 | + 2.82·7-s − 3·9-s − 6.32·11-s + 4.47·13-s − 6.32·19-s + 8.48·23-s − 4.47·37-s + 2·41-s − 2.82·47-s + 1.00·49-s + 13.4·53-s + 6.32·59-s − 8.48·63-s − 17.8·77-s + 9·81-s + 14·89-s + 12.6·91-s + 18.9·99-s + 19.7·103-s − 13.4·117-s + ⋯ |
| L(s) = 1 | + 1.06·7-s − 9-s − 1.90·11-s + 1.24·13-s − 1.45·19-s + 1.76·23-s − 0.735·37-s + 0.312·41-s − 0.412·47-s + 0.142·49-s + 1.84·53-s + 0.823·59-s − 1.06·63-s − 2.03·77-s + 81-s + 1.48·89-s + 1.32·91-s + 1.90·99-s + 1.95·103-s − 1.24·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.717792583\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.717792583\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 3T^{2} \) |
| 7 | \( 1 - 2.82T + 7T^{2} \) |
| 11 | \( 1 + 6.32T + 11T^{2} \) |
| 13 | \( 1 - 4.47T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 6.32T + 19T^{2} \) |
| 23 | \( 1 - 8.48T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 4.47T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 2.82T + 47T^{2} \) |
| 53 | \( 1 - 13.4T + 53T^{2} \) |
| 59 | \( 1 - 6.32T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 14T + 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−8.194053751078541039485379451500, −7.46653989479500886474975013995, −6.59524664052081623388234805279, −5.72690018408399215610948321705, −5.21616069665935701661215643404, −4.59507693393422520929209621113, −3.52413751406553569516076847105, −2.68938403856704086596122066369, −1.95705241053648723725728959061, −0.66484135115214633738124175840,
0.66484135115214633738124175840, 1.95705241053648723725728959061, 2.68938403856704086596122066369, 3.52413751406553569516076847105, 4.59507693393422520929209621113, 5.21616069665935701661215643404, 5.72690018408399215610948321705, 6.59524664052081623388234805279, 7.46653989479500886474975013995, 8.194053751078541039485379451500
