L(s) = 1 | − 3-s − 0.561·5-s − 3.56·7-s + 9-s − 2·11-s + 0.561·15-s + 2.56·17-s − 1.12·19-s + 3.56·21-s − 2·23-s − 4.68·25-s − 27-s − 5.68·29-s − 1.56·31-s + 2·33-s + 2·35-s − 3.43·37-s − 2.56·41-s − 0.438·43-s − 0.561·45-s − 8.24·47-s + 5.68·49-s − 2.56·51-s + 11.6·53-s + 1.12·55-s + 1.12·57-s − 11.1·59-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.251·5-s − 1.34·7-s + 0.333·9-s − 0.603·11-s + 0.144·15-s + 0.621·17-s − 0.257·19-s + 0.777·21-s − 0.417·23-s − 0.936·25-s − 0.192·27-s − 1.05·29-s − 0.280·31-s + 0.348·33-s + 0.338·35-s − 0.565·37-s − 0.400·41-s − 0.0668·43-s − 0.0837·45-s − 1.20·47-s + 0.812·49-s − 0.358·51-s + 1.60·53-s + 0.151·55-s + 0.148·57-s − 1.44·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4855054791\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4855054791\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 + 0.561T + 5T^{2} \) |
| 7 | \( 1 + 3.56T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 17 | \( 1 - 2.56T + 17T^{2} \) |
| 19 | \( 1 + 1.12T + 19T^{2} \) |
| 23 | \( 1 + 2T + 23T^{2} \) |
| 29 | \( 1 + 5.68T + 29T^{2} \) |
| 31 | \( 1 + 1.56T + 31T^{2} \) |
| 37 | \( 1 + 3.43T + 37T^{2} \) |
| 41 | \( 1 + 2.56T + 41T^{2} \) |
| 43 | \( 1 + 0.438T + 43T^{2} \) |
| 47 | \( 1 + 8.24T + 47T^{2} \) |
| 53 | \( 1 - 11.6T + 53T^{2} \) |
| 59 | \( 1 + 11.1T + 59T^{2} \) |
| 61 | \( 1 - 12.1T + 61T^{2} \) |
| 67 | \( 1 - 0.438T + 67T^{2} \) |
| 71 | \( 1 - 14T + 71T^{2} \) |
| 73 | \( 1 - 1.87T + 73T^{2} \) |
| 79 | \( 1 + 9.56T + 79T^{2} \) |
| 83 | \( 1 + 9.12T + 83T^{2} \) |
| 89 | \( 1 + 13.1T + 89T^{2} \) |
| 97 | \( 1 - 4.43T + 97T^{2} \) |
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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
−7.75989882426530774426488202845, −7.00574725072672646084387518420, −6.49111584323915903448006788378, −5.65127646323977032447569880297, −5.30857740717724732251103929269, −4.12486866170617628799417483280, −3.60434585419309175848287516954, −2.76627999624588944411643042959, −1.72440110865859184768852377500, −0.34215454935970406850878743825,
0.34215454935970406850878743825, 1.72440110865859184768852377500, 2.76627999624588944411643042959, 3.60434585419309175848287516954, 4.12486866170617628799417483280, 5.30857740717724732251103929269, 5.65127646323977032447569880297, 6.49111584323915903448006788378, 7.00574725072672646084387518420, 7.75989882426530774426488202845