| L(s) = 1 | − 7·3-s + 16·4-s − 32·9-s + 121·11-s − 112·12-s + 256·16-s − 167·23-s + 791·27-s − 553·31-s − 847·33-s − 512·36-s + 2.11e3·37-s + 1.93e3·44-s + 1.91e3·47-s − 1.79e3·48-s + 2.40e3·49-s + 718·53-s + 4.48e3·59-s + 4.09e3·64-s + 7.75e3·67-s + 1.16e3·69-s + 7.60e3·71-s − 2.94e3·81-s − 6.43e3·89-s − 2.67e3·92-s + 3.87e3·93-s + 9.79e3·97-s + ⋯ |
| L(s) = 1 | − 7/9·3-s + 4-s − 0.395·9-s + 11-s − 7/9·12-s + 16-s − 0.315·23-s + 1.08·27-s − 0.575·31-s − 7/9·33-s − 0.395·36-s + 1.54·37-s + 44-s + 0.868·47-s − 7/9·48-s + 49-s + 0.255·53-s + 1.28·59-s + 64-s + 1.72·67-s + 0.245·69-s + 1.50·71-s − 0.448·81-s − 0.812·89-s − 0.315·92-s + 0.447·93-s + 1.04·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.917851695\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.917851695\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 - p^{2} T \) |
| good | 2 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 3 | \( 1 + 7 T + p^{4} T^{2} \) |
| 7 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 13 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 17 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 19 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 23 | \( 1 + 167 T + p^{4} T^{2} \) |
| 29 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 31 | \( 1 + 553 T + p^{4} T^{2} \) |
| 37 | \( 1 - 2113 T + p^{4} T^{2} \) |
| 41 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 43 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 47 | \( 1 - 1918 T + p^{4} T^{2} \) |
| 53 | \( 1 - 718 T + p^{4} T^{2} \) |
| 59 | \( 1 - 4487 T + p^{4} T^{2} \) |
| 61 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 67 | \( 1 - 7753 T + p^{4} T^{2} \) |
| 71 | \( 1 - 7607 T + p^{4} T^{2} \) |
| 73 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 79 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 83 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 89 | \( 1 + 6433 T + p^{4} T^{2} \) |
| 97 | \( 1 - 9793 T + p^{4} T^{2} \) |
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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
−11.35638834289702611147438170237, −10.59503317237476959971244725426, −9.466399368896240202751142479489, −8.246877520358575604033153247867, −7.05540670263739542606916631594, −6.25567904650276660026179649655, −5.43000366258736901657075423445, −3.87385216770516536395146827868, −2.42309812595471065722378354206, −0.925741555458872153331813960097,
0.925741555458872153331813960097, 2.42309812595471065722378354206, 3.87385216770516536395146827868, 5.43000366258736901657075423445, 6.25567904650276660026179649655, 7.05540670263739542606916631594, 8.246877520358575604033153247867, 9.466399368896240202751142479489, 10.59503317237476959971244725426, 11.35638834289702611147438170237
