| L(s) = 1 | + 3-s − 4.60·7-s + 9-s + 11-s + 4.60·13-s − 6.60·17-s − 7.21·19-s − 4.60·21-s + 27-s + 8·29-s + 9.21·31-s + 33-s + 3.21·37-s + 4.60·39-s + 8·41-s + 3.39·43-s + 5.21·47-s + 14.2·49-s − 6.60·51-s − 2·53-s − 7.21·57-s + 8·59-s + 7.21·61-s − 4.60·63-s + 4·67-s − 14.4·71-s − 0.605·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.74·7-s + 0.333·9-s + 0.301·11-s + 1.27·13-s − 1.60·17-s − 1.65·19-s − 1.00·21-s + 0.192·27-s + 1.48·29-s + 1.65·31-s + 0.174·33-s + 0.527·37-s + 0.737·39-s + 1.24·41-s + 0.517·43-s + 0.760·47-s + 2.03·49-s − 0.924·51-s − 0.274·53-s − 0.955·57-s + 1.04·59-s + 0.923·61-s − 0.580·63-s + 0.488·67-s − 1.71·71-s − 0.0708·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.773045704\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.773045704\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 7 | \( 1 + 4.60T + 7T^{2} \) |
| 13 | \( 1 - 4.60T + 13T^{2} \) |
| 17 | \( 1 + 6.60T + 17T^{2} \) |
| 19 | \( 1 + 7.21T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 8T + 29T^{2} \) |
| 31 | \( 1 - 9.21T + 31T^{2} \) |
| 37 | \( 1 - 3.21T + 37T^{2} \) |
| 41 | \( 1 - 8T + 41T^{2} \) |
| 43 | \( 1 - 3.39T + 43T^{2} \) |
| 47 | \( 1 - 5.21T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 - 8T + 59T^{2} \) |
| 61 | \( 1 - 7.21T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 14.4T + 71T^{2} \) |
| 73 | \( 1 + 0.605T + 73T^{2} \) |
| 79 | \( 1 + 11.2T + 79T^{2} \) |
| 83 | \( 1 - 10.6T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 12.4T + 97T^{2} \) |
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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
−8.813916005310831480734545076274, −8.104871630864061629619481808515, −6.85971671724667968473629109763, −6.45434669837707989332021700025, −6.00477281524175025567788726305, −4.37912285316047096388157517334, −4.02758490704729885785208751627, −2.95083270163911374555007857731, −2.32153715271758619603875554053, −0.75645246994083338533263384375,
0.75645246994083338533263384375, 2.32153715271758619603875554053, 2.95083270163911374555007857731, 4.02758490704729885785208751627, 4.37912285316047096388157517334, 6.00477281524175025567788726305, 6.45434669837707989332021700025, 6.85971671724667968473629109763, 8.104871630864061629619481808515, 8.813916005310831480734545076274
