| L(s) = 1 | − 2·3-s + 5-s − 7-s + 9-s + 4·11-s − 2·15-s + 6·19-s + 2·21-s + 2·23-s + 25-s + 4·27-s − 6·29-s − 8·31-s − 8·33-s − 35-s − 6·37-s + 8·41-s + 4·43-s + 45-s − 8·47-s + 49-s + 4·55-s − 12·57-s + 10·59-s + 14·61-s − 63-s + 4·67-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s + 1.20·11-s − 0.516·15-s + 1.37·19-s + 0.436·21-s + 0.417·23-s + 1/5·25-s + 0.769·27-s − 1.11·29-s − 1.43·31-s − 1.39·33-s − 0.169·35-s − 0.986·37-s + 1.24·41-s + 0.609·43-s + 0.149·45-s − 1.16·47-s + 1/7·49-s + 0.539·55-s − 1.58·57-s + 1.30·59-s + 1.79·61-s − 0.125·63-s + 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−12.60719046790795, −12.24814432824087, −11.66045037675737, −11.38306633847351, −11.06400397267836, −10.56343911262701, −9.921640503474118, −9.590345463147823, −9.229598467415355, −8.733097483585017, −8.220059675544093, −7.332954948674060, −7.148139096748113, −6.700052882495398, −6.147476554057261, −5.680925220649372, −5.360350681106295, −5.012736536300966, −4.143963313071947, −3.800208238142297, −3.229719387254426, −2.600391972336462, −1.840359371811226, −1.275413032507990, −0.7561155381435207, 0,
0.7561155381435207, 1.275413032507990, 1.840359371811226, 2.600391972336462, 3.229719387254426, 3.800208238142297, 4.143963313071947, 5.012736536300966, 5.360350681106295, 5.680925220649372, 6.147476554057261, 6.700052882495398, 7.148139096748113, 7.332954948674060, 8.220059675544093, 8.733097483585017, 9.229598467415355, 9.590345463147823, 9.921640503474118, 10.56343911262701, 11.06400397267836, 11.38306633847351, 11.66045037675737, 12.24814432824087, 12.60719046790795
