L(s) = 1 | + 2.93·3-s + 3.77·5-s + 2.25·7-s + 5.59·9-s − 11-s − 1.48·13-s + 11.0·15-s + 3.18·17-s + 19-s + 6.61·21-s − 8.68·23-s + 9.26·25-s + 7.60·27-s − 9.28·29-s + 3.52·31-s − 2.93·33-s + 8.51·35-s + 8.84·37-s − 4.34·39-s − 10.9·41-s − 5.47·43-s + 21.1·45-s + 3.55·47-s − 1.91·49-s + 9.35·51-s − 0.536·53-s − 3.77·55-s + ⋯ |
L(s) = 1 | + 1.69·3-s + 1.68·5-s + 0.852·7-s + 1.86·9-s − 0.301·11-s − 0.411·13-s + 2.85·15-s + 0.773·17-s + 0.229·19-s + 1.44·21-s − 1.81·23-s + 1.85·25-s + 1.46·27-s − 1.72·29-s + 0.633·31-s − 0.510·33-s + 1.43·35-s + 1.45·37-s − 0.695·39-s − 1.70·41-s − 0.834·43-s + 3.14·45-s + 0.518·47-s − 0.273·49-s + 1.30·51-s − 0.0736·53-s − 0.509·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.175029848\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.175029848\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 - 2.93T + 3T^{2} \) |
| 5 | \( 1 - 3.77T + 5T^{2} \) |
| 7 | \( 1 - 2.25T + 7T^{2} \) |
| 13 | \( 1 + 1.48T + 13T^{2} \) |
| 17 | \( 1 - 3.18T + 17T^{2} \) |
| 23 | \( 1 + 8.68T + 23T^{2} \) |
| 29 | \( 1 + 9.28T + 29T^{2} \) |
| 31 | \( 1 - 3.52T + 31T^{2} \) |
| 37 | \( 1 - 8.84T + 37T^{2} \) |
| 41 | \( 1 + 10.9T + 41T^{2} \) |
| 43 | \( 1 + 5.47T + 43T^{2} \) |
| 47 | \( 1 - 3.55T + 47T^{2} \) |
| 53 | \( 1 + 0.536T + 53T^{2} \) |
| 59 | \( 1 + 13.7T + 59T^{2} \) |
| 61 | \( 1 - 8.03T + 61T^{2} \) |
| 67 | \( 1 - 8.22T + 67T^{2} \) |
| 71 | \( 1 + 6.42T + 71T^{2} \) |
| 73 | \( 1 - 1.66T + 73T^{2} \) |
| 79 | \( 1 + 12.0T + 79T^{2} \) |
| 83 | \( 1 - 5.79T + 83T^{2} \) |
| 89 | \( 1 - 3.88T + 89T^{2} \) |
| 97 | \( 1 - 15.5T + 97T^{2} \) |
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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
−8.622327681096374000919671255695, −7.945571525261231418028970224958, −7.47688755413831939090196755028, −6.36368814439777764585181785488, −5.56774280073741397450985089912, −4.78816237440784302655237820112, −3.74887031567320762677097148995, −2.80090459947056070764888378042, −2.02157741303540660339650693164, −1.56846475104487573032851499350,
1.56846475104487573032851499350, 2.02157741303540660339650693164, 2.80090459947056070764888378042, 3.74887031567320762677097148995, 4.78816237440784302655237820112, 5.56774280073741397450985089912, 6.36368814439777764585181785488, 7.47688755413831939090196755028, 7.945571525261231418028970224958, 8.622327681096374000919671255695