| L(s) = 1 | − 3·3-s + 4·4-s + 5·5-s + 9·9-s + 7·11-s − 12·12-s − 15·15-s + 16·16-s + 20·20-s − 41·23-s + 25·25-s − 27·27-s + 29·29-s − 21·33-s + 36·36-s + 71·37-s − 53·41-s + 59·43-s + 28·44-s + 45·45-s − 48·48-s + 49·49-s + 19·53-s + 35·55-s − 60·60-s + 64·64-s + 123·69-s + ⋯ |
| L(s) = 1 | − 3-s + 4-s + 5-s + 9-s + 7/11·11-s − 12-s − 15-s + 16-s + 20-s − 1.78·23-s + 25-s − 27-s + 29-s − 0.636·33-s + 36-s + 1.91·37-s − 1.29·41-s + 1.37·43-s + 7/11·44-s + 45-s − 48-s + 49-s + 0.358·53-s + 7/11·55-s − 60-s + 64-s + 1.78·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.012013371\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.012013371\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + p T \) |
| 5 | \( 1 - p T \) |
| 29 | \( 1 - p T \) |
| good | 2 | \( ( 1 - p T )( 1 + p T ) \) |
| 7 | \( ( 1 - p T )( 1 + p T ) \) |
| 11 | \( 1 - 7 T + p^{2} T^{2} \) |
| 13 | \( ( 1 - p T )( 1 + p T ) \) |
| 17 | \( ( 1 - p T )( 1 + p T ) \) |
| 19 | \( ( 1 - p T )( 1 + p T ) \) |
| 23 | \( 1 + 41 T + p^{2} T^{2} \) |
| 31 | \( ( 1 - p T )( 1 + p T ) \) |
| 37 | \( 1 - 71 T + p^{2} T^{2} \) |
| 41 | \( 1 + 53 T + p^{2} T^{2} \) |
| 43 | \( 1 - 59 T + p^{2} T^{2} \) |
| 47 | \( ( 1 - p T )( 1 + p T ) \) |
| 53 | \( 1 - 19 T + p^{2} T^{2} \) |
| 59 | \( ( 1 - p T )( 1 + p T ) \) |
| 61 | \( ( 1 - p T )( 1 + p T ) \) |
| 67 | \( ( 1 - p T )( 1 + p T ) \) |
| 71 | \( ( 1 - p T )( 1 + p T ) \) |
| 73 | \( 1 + T + p^{2} T^{2} \) |
| 79 | \( ( 1 - p T )( 1 + p T ) \) |
| 83 | \( 1 - 79 T + p^{2} T^{2} \) |
| 89 | \( 1 + 62 T + p^{2} T^{2} \) |
| 97 | \( 1 + 49 T + p^{2} T^{2} \) |
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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
−10.88965458956183607328413247932, −10.20844562524526650929498218394, −9.468201675371588189844156313502, −8.002005883923759507259649282931, −6.84859645840711605667824252059, −6.21015608819662411179756023276, −5.53425175704503221255828643468, −4.14738178195932642111416106758, −2.42269363291139208172532716134, −1.23528383259415825245829677177,
1.23528383259415825245829677177, 2.42269363291139208172532716134, 4.14738178195932642111416106758, 5.53425175704503221255828643468, 6.21015608819662411179756023276, 6.84859645840711605667824252059, 8.002005883923759507259649282931, 9.468201675371588189844156313502, 10.20844562524526650929498218394, 10.88965458956183607328413247932
