| L(s) = 1 | + 5·5-s + 18.8·7-s + 63.2·11-s + 1.58·13-s − 135.·17-s + 97.2·19-s − 41.1·23-s + 25·25-s + 207.·29-s + 193.·31-s + 94.0·35-s + 339.·37-s + 490.·41-s + 74.3·43-s − 544.·47-s + 10.6·49-s − 663.·53-s + 316.·55-s − 344.·59-s + 5.16·61-s + 7.90·65-s − 671.·67-s − 425.·71-s − 94.8·73-s + 1.18e3·77-s + 770.·79-s − 589.·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.01·7-s + 1.73·11-s + 0.0337·13-s − 1.92·17-s + 1.17·19-s − 0.373·23-s + 0.200·25-s + 1.32·29-s + 1.12·31-s + 0.454·35-s + 1.50·37-s + 1.86·41-s + 0.263·43-s − 1.68·47-s + 0.0311·49-s − 1.72·53-s + 0.775·55-s − 0.760·59-s + 0.0108·61-s + 0.0150·65-s − 1.22·67-s − 0.711·71-s − 0.152·73-s + 1.75·77-s + 1.09·79-s − 0.779·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.223243503\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.223243503\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| good | 7 | \( 1 - 18.8T + 343T^{2} \) |
| 11 | \( 1 - 63.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 1.58T + 2.19e3T^{2} \) |
| 17 | \( 1 + 135.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 97.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 41.1T + 1.21e4T^{2} \) |
| 29 | \( 1 - 207.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 193.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 339.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 490.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 74.3T + 7.95e4T^{2} \) |
| 47 | \( 1 + 544.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 663.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 344.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 5.16T + 2.26e5T^{2} \) |
| 67 | \( 1 + 671.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 425.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 94.8T + 3.89e5T^{2} \) |
| 79 | \( 1 - 770.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 589.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 409.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 152.T + 9.12e5T^{2} \) |
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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
−9.213595244391934959446048557327, −8.436922033786721115134444438301, −7.60242446095971897748870746228, −6.50944910901351755425314052917, −6.12554095848348837753589562827, −4.66357336385897020056234615627, −4.39301499730362954142711136343, −2.95357360607826357190807246409, −1.80742500223064595230262743908, −0.960886566648289208775913192658,
0.960886566648289208775913192658, 1.80742500223064595230262743908, 2.95357360607826357190807246409, 4.39301499730362954142711136343, 4.66357336385897020056234615627, 6.12554095848348837753589562827, 6.50944910901351755425314052917, 7.60242446095971897748870746228, 8.436922033786721115134444438301, 9.213595244391934959446048557327
