| L(s) = 1 | − 2-s + 3-s + 4-s + 0.686·5-s − 6-s + 2.87·7-s − 8-s + 9-s − 0.686·10-s + 5.51·11-s + 12-s + 0.329·13-s − 2.87·14-s + 0.686·15-s + 16-s + 4.44·17-s − 18-s − 1.53·19-s + 0.686·20-s + 2.87·21-s − 5.51·22-s − 3.17·23-s − 24-s − 4.52·25-s − 0.329·26-s + 27-s + 2.87·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.306·5-s − 0.408·6-s + 1.08·7-s − 0.353·8-s + 0.333·9-s − 0.217·10-s + 1.66·11-s + 0.288·12-s + 0.0914·13-s − 0.768·14-s + 0.177·15-s + 0.250·16-s + 1.07·17-s − 0.235·18-s − 0.352·19-s + 0.153·20-s + 0.627·21-s − 1.17·22-s − 0.661·23-s − 0.204·24-s − 0.905·25-s − 0.0646·26-s + 0.192·27-s + 0.543·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1338 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.991333424\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.991333424\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 223 | \( 1 + T \) |
| good | 5 | \( 1 - 0.686T + 5T^{2} \) |
| 7 | \( 1 - 2.87T + 7T^{2} \) |
| 11 | \( 1 - 5.51T + 11T^{2} \) |
| 13 | \( 1 - 0.329T + 13T^{2} \) |
| 17 | \( 1 - 4.44T + 17T^{2} \) |
| 19 | \( 1 + 1.53T + 19T^{2} \) |
| 23 | \( 1 + 3.17T + 23T^{2} \) |
| 29 | \( 1 - 7.03T + 29T^{2} \) |
| 31 | \( 1 + 7.43T + 31T^{2} \) |
| 37 | \( 1 + 7.58T + 37T^{2} \) |
| 41 | \( 1 + 2.51T + 41T^{2} \) |
| 43 | \( 1 - 6.97T + 43T^{2} \) |
| 47 | \( 1 - 6.00T + 47T^{2} \) |
| 53 | \( 1 + 0.840T + 53T^{2} \) |
| 59 | \( 1 + 3.14T + 59T^{2} \) |
| 61 | \( 1 + 2.68T + 61T^{2} \) |
| 67 | \( 1 + 13.9T + 67T^{2} \) |
| 71 | \( 1 + 6.67T + 71T^{2} \) |
| 73 | \( 1 - 11.9T + 73T^{2} \) |
| 79 | \( 1 - 12.1T + 79T^{2} \) |
| 83 | \( 1 + 5.22T + 83T^{2} \) |
| 89 | \( 1 - 8.29T + 89T^{2} \) |
| 97 | \( 1 - 4.45T + 97T^{2} \) |
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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
−9.419112382552145271797561204432, −8.871491837523768758035389703892, −8.096975643766720450402313100290, −7.42402003461030987107746378299, −6.47939684130714623645796745259, −5.58394223470196245039365133389, −4.34111949263638118387840789247, −3.45149027049067907494587218443, −2.01452456951285869762546390411, −1.29373614850023603182575226277,
1.29373614850023603182575226277, 2.01452456951285869762546390411, 3.45149027049067907494587218443, 4.34111949263638118387840789247, 5.58394223470196245039365133389, 6.47939684130714623645796745259, 7.42402003461030987107746378299, 8.096975643766720450402313100290, 8.871491837523768758035389703892, 9.419112382552145271797561204432
