| L(s) = 1 | + 3·3-s − 0.726·5-s + 9·9-s + 64.4·11-s − 71.8·13-s − 2.17·15-s − 48.9·17-s + 34.3·19-s + 0.903·23-s − 124.·25-s + 27·27-s + 226.·29-s − 275.·31-s + 193.·33-s + 295.·37-s − 215.·39-s − 186.·41-s + 455.·43-s − 6.53·45-s + 282.·47-s − 146.·51-s + 356.·53-s − 46.8·55-s + 103.·57-s + 729.·59-s − 274.·61-s + 52.1·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.0649·5-s + 0.333·9-s + 1.76·11-s − 1.53·13-s − 0.0375·15-s − 0.697·17-s + 0.415·19-s + 0.00818·23-s − 0.995·25-s + 0.192·27-s + 1.45·29-s − 1.59·31-s + 1.02·33-s + 1.31·37-s − 0.884·39-s − 0.710·41-s + 1.61·43-s − 0.0216·45-s + 0.876·47-s − 0.402·51-s + 0.923·53-s − 0.114·55-s + 0.239·57-s + 1.60·59-s − 0.575·61-s + 0.0995·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.838445650\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.838445650\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 0.726T + 125T^{2} \) |
| 11 | \( 1 - 64.4T + 1.33e3T^{2} \) |
| 13 | \( 1 + 71.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 48.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 34.3T + 6.85e3T^{2} \) |
| 23 | \( 1 - 0.903T + 1.21e4T^{2} \) |
| 29 | \( 1 - 226.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 275.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 295.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 186.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 455.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 282.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 356.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 729.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 274.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 193.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 40.5T + 3.57e5T^{2} \) |
| 73 | \( 1 - 206.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 937.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 911.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 949.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 39.4T + 9.12e5T^{2} \) |
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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
−8.824640394826639948530446460119, −7.78388801512198031749572621543, −7.15946374303968577534339029030, −6.48072152840715334481716554887, −5.47753945225793074171702242928, −4.38766330391293568108166617855, −3.89600502488203313208177934209, −2.73299720152024619801286306303, −1.90780736966263454895888517634, −0.73130621586554670798405678225,
0.73130621586554670798405678225, 1.90780736966263454895888517634, 2.73299720152024619801286306303, 3.89600502488203313208177934209, 4.38766330391293568108166617855, 5.47753945225793074171702242928, 6.48072152840715334481716554887, 7.15946374303968577534339029030, 7.78388801512198031749572621543, 8.824640394826639948530446460119
