| L(s) = 1 | + 3·3-s + 14.3·5-s − 3.76·7-s + 9·9-s − 34.4·13-s + 43.0·15-s − 45.4·17-s + 21.9·19-s − 11.2·21-s − 0.925·23-s + 81.1·25-s + 27·27-s + 34.8·29-s + 186.·31-s − 54.0·35-s + 295.·37-s − 103.·39-s + 341.·41-s − 6.89·43-s + 129.·45-s + 307.·47-s − 328.·49-s − 136.·51-s + 480.·53-s + 65.7·57-s + 444.·59-s + 530.·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.28·5-s − 0.203·7-s + 0.333·9-s − 0.734·13-s + 0.741·15-s − 0.648·17-s + 0.264·19-s − 0.117·21-s − 0.00838·23-s + 0.648·25-s + 0.192·27-s + 0.223·29-s + 1.07·31-s − 0.261·35-s + 1.31·37-s − 0.424·39-s + 1.30·41-s − 0.0244·43-s + 0.428·45-s + 0.954·47-s − 0.958·49-s − 0.374·51-s + 1.24·53-s + 0.152·57-s + 0.981·59-s + 1.11·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.479539865\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.479539865\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 - 14.3T + 125T^{2} \) |
| 7 | \( 1 + 3.76T + 343T^{2} \) |
| 13 | \( 1 + 34.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 45.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 21.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 0.925T + 1.21e4T^{2} \) |
| 29 | \( 1 - 34.8T + 2.43e4T^{2} \) |
| 31 | \( 1 - 186.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 295.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 341.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 6.89T + 7.95e4T^{2} \) |
| 47 | \( 1 - 307.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 480.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 444.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 530.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 227.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 464.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 421.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 893.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 409.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 908.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 437.T + 9.12e5T^{2} \) |
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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
−9.320382381421025862869987006457, −8.472495924794470392052621059607, −7.54833351138109380414270887771, −6.66601825286706087297255843867, −5.93017194697858350348087407599, −5.00400297251129831412999300244, −4.04733803716166598797532412680, −2.70088242064609160486526761069, −2.21036550700652578446556782613, −0.905188935480392563796496788606,
0.905188935480392563796496788606, 2.21036550700652578446556782613, 2.70088242064609160486526761069, 4.04733803716166598797532412680, 5.00400297251129831412999300244, 5.93017194697858350348087407599, 6.66601825286706087297255843867, 7.54833351138109380414270887771, 8.472495924794470392052621059607, 9.320382381421025862869987006457
