| L(s) = 1 | − 3·3-s − 5-s + 3·7-s + 6·9-s − 5·11-s + 2·13-s + 3·15-s + 4·17-s + 4·19-s − 9·21-s − 6·23-s + 25-s − 9·27-s + 6·29-s + 4·31-s + 15·33-s − 3·35-s − 37-s − 6·39-s − 9·41-s − 10·43-s − 6·45-s + 11·47-s + 2·49-s − 12·51-s − 11·53-s + 5·55-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 0.447·5-s + 1.13·7-s + 2·9-s − 1.50·11-s + 0.554·13-s + 0.774·15-s + 0.970·17-s + 0.917·19-s − 1.96·21-s − 1.25·23-s + 1/5·25-s − 1.73·27-s + 1.11·29-s + 0.718·31-s + 2.61·33-s − 0.507·35-s − 0.164·37-s − 0.960·39-s − 1.40·41-s − 1.52·43-s − 0.894·45-s + 1.60·47-s + 2/7·49-s − 1.68·51-s − 1.51·53-s + 0.674·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8826363900\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8826363900\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 37 | \( 1 + T \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 11 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−8.384495914592605943198193585522, −7.967716714125813453392078887841, −7.23620303213902323108245082851, −6.31278841946632243083099496990, −5.49557340632345683992432020156, −5.06756007617691541853072307264, −4.40343244869593798657772342377, −3.20508638243343033078558974345, −1.70436716945358454725317464833, −0.65053381069411396111812346191,
0.65053381069411396111812346191, 1.70436716945358454725317464833, 3.20508638243343033078558974345, 4.40343244869593798657772342377, 5.06756007617691541853072307264, 5.49557340632345683992432020156, 6.31278841946632243083099496990, 7.23620303213902323108245082851, 7.967716714125813453392078887841, 8.384495914592605943198193585522
