| L(s) = 1 | − 9·3-s + 244·7-s + 81·9-s − 144·11-s − 50·13-s + 1.91e3·17-s + 140·19-s − 2.19e3·21-s + 624·23-s − 729·27-s − 3.12e3·29-s − 5.17e3·31-s + 1.29e3·33-s − 1.56e4·37-s + 450·39-s + 1.25e4·41-s − 1.15e4·43-s + 2.67e4·47-s + 4.27e4·49-s − 1.72e4·51-s + 1.91e4·53-s − 1.26e3·57-s + 2.79e4·59-s + 2.20e4·61-s + 1.97e4·63-s + 1.26e4·67-s − 5.61e3·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.88·7-s + 1/3·9-s − 0.358·11-s − 0.0820·13-s + 1.60·17-s + 0.0889·19-s − 1.08·21-s + 0.245·23-s − 0.192·27-s − 0.690·29-s − 0.967·31-s + 0.207·33-s − 1.88·37-s + 0.0473·39-s + 1.16·41-s − 0.949·43-s + 1.76·47-s + 2.54·49-s − 0.927·51-s + 0.936·53-s − 0.0513·57-s + 1.04·59-s + 0.757·61-s + 0.627·63-s + 0.344·67-s − 0.142·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.203502609\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.203502609\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p^{2} T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 244 T + p^{5} T^{2} \) |
| 11 | \( 1 + 144 T + p^{5} T^{2} \) |
| 13 | \( 1 + 50 T + p^{5} T^{2} \) |
| 17 | \( 1 - 1914 T + p^{5} T^{2} \) |
| 19 | \( 1 - 140 T + p^{5} T^{2} \) |
| 23 | \( 1 - 624 T + p^{5} T^{2} \) |
| 29 | \( 1 + 3126 T + p^{5} T^{2} \) |
| 31 | \( 1 + 5176 T + p^{5} T^{2} \) |
| 37 | \( 1 + 15698 T + p^{5} T^{2} \) |
| 41 | \( 1 - 12570 T + p^{5} T^{2} \) |
| 43 | \( 1 + 11516 T + p^{5} T^{2} \) |
| 47 | \( 1 - 26736 T + p^{5} T^{2} \) |
| 53 | \( 1 - 19158 T + p^{5} T^{2} \) |
| 59 | \( 1 - 27984 T + p^{5} T^{2} \) |
| 61 | \( 1 - 22022 T + p^{5} T^{2} \) |
| 67 | \( 1 - 12676 T + p^{5} T^{2} \) |
| 71 | \( 1 + 59520 T + p^{5} T^{2} \) |
| 73 | \( 1 - 67102 T + p^{5} T^{2} \) |
| 79 | \( 1 - 11048 T + p^{5} T^{2} \) |
| 83 | \( 1 - 115284 T + p^{5} T^{2} \) |
| 89 | \( 1 - 73650 T + p^{5} T^{2} \) |
| 97 | \( 1 + 35522 T + p^{5} T^{2} \) |
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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
−10.93648210683167849255548714160, −10.24983631817767854359640325186, −8.914535246675628518701448090550, −7.88116188327387936077369607471, −7.22876705413046419750484474001, −5.54500228013529253037622932987, −5.13141525974496658561464644957, −3.81078780079936262133727673200, −2.01725202827975941600263018333, −0.913490966574613453564506930109,
0.913490966574613453564506930109, 2.01725202827975941600263018333, 3.81078780079936262133727673200, 5.13141525974496658561464644957, 5.54500228013529253037622932987, 7.22876705413046419750484474001, 7.88116188327387936077369607471, 8.914535246675628518701448090550, 10.24983631817767854359640325186, 10.93648210683167849255548714160
