| L(s) = 1 | − 3·3-s − 5·5-s + 8.04·7-s + 9·9-s − 31.4·11-s − 13·13-s + 15·15-s + 32.7·17-s + 131.·19-s − 24.1·21-s − 175.·23-s + 25·25-s − 27·27-s − 96.4·29-s + 146.·31-s + 94.4·33-s − 40.2·35-s + 10.6·37-s + 39·39-s + 159.·41-s + 216.·43-s − 45·45-s − 16.2·47-s − 278.·49-s − 98.3·51-s + 181.·53-s + 157.·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.434·7-s + 0.333·9-s − 0.862·11-s − 0.277·13-s + 0.258·15-s + 0.467·17-s + 1.59·19-s − 0.250·21-s − 1.58·23-s + 0.200·25-s − 0.192·27-s − 0.617·29-s + 0.851·31-s + 0.498·33-s − 0.194·35-s + 0.0471·37-s + 0.160·39-s + 0.605·41-s + 0.767·43-s − 0.149·45-s − 0.0503·47-s − 0.811·49-s − 0.270·51-s + 0.470·53-s + 0.385·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 5 | \( 1 + 5T \) |
| 13 | \( 1 + 13T \) |
| good | 7 | \( 1 - 8.04T + 343T^{2} \) |
| 11 | \( 1 + 31.4T + 1.33e3T^{2} \) |
| 17 | \( 1 - 32.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 131.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 175.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 96.4T + 2.43e4T^{2} \) |
| 31 | \( 1 - 146.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 10.6T + 5.06e4T^{2} \) |
| 41 | \( 1 - 159.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 216.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 16.2T + 1.03e5T^{2} \) |
| 53 | \( 1 - 181.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 614.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 712.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 696.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 961.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.07e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 922.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.22e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 433.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 774.T + 9.12e5T^{2} \) |
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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
−8.502742602759030715754529184638, −7.72338142838901467487529166381, −7.30202169113484156023963691043, −6.05388071744245288702678707126, −5.38280416370255515544702380052, −4.59279214085267031994883762332, −3.59879317561559448501505582323, −2.46198436323207998472882072740, −1.15586802919173863936567824391, 0,
1.15586802919173863936567824391, 2.46198436323207998472882072740, 3.59879317561559448501505582323, 4.59279214085267031994883762332, 5.38280416370255515544702380052, 6.05388071744245288702678707126, 7.30202169113484156023963691043, 7.72338142838901467487529166381, 8.502742602759030715754529184638
