L(s) = 1 | + 0.618·2-s + 3-s − 1.61·4-s + 0.618·6-s − 0.763·7-s − 2.23·8-s + 9-s − 1.61·12-s − 1.23·13-s − 0.472·14-s + 1.85·16-s + 2·17-s + 0.618·18-s − 5·19-s − 0.763·21-s − 2.38·23-s − 2.23·24-s − 0.763·26-s + 27-s + 1.23·28-s + 5.85·29-s + 7·31-s + 5.61·32-s + 1.23·34-s − 1.61·36-s + 7.47·37-s − 3.09·38-s + ⋯ |
L(s) = 1 | + 0.437·2-s + 0.577·3-s − 0.809·4-s + 0.252·6-s − 0.288·7-s − 0.790·8-s + 0.333·9-s − 0.467·12-s − 0.342·13-s − 0.126·14-s + 0.463·16-s + 0.485·17-s + 0.145·18-s − 1.14·19-s − 0.166·21-s − 0.496·23-s − 0.456·24-s − 0.149·26-s + 0.192·27-s + 0.233·28-s + 1.08·29-s + 1.25·31-s + 0.993·32-s + 0.211·34-s − 0.269·36-s + 1.22·37-s − 0.501·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - 0.618T + 2T^{2} \) |
| 7 | \( 1 + 0.763T + 7T^{2} \) |
| 13 | \( 1 + 1.23T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 + 5T + 19T^{2} \) |
| 23 | \( 1 + 2.38T + 23T^{2} \) |
| 29 | \( 1 - 5.85T + 29T^{2} \) |
| 31 | \( 1 - 7T + 31T^{2} \) |
| 37 | \( 1 - 7.47T + 37T^{2} \) |
| 41 | \( 1 + 2.52T + 41T^{2} \) |
| 43 | \( 1 + 2.09T + 43T^{2} \) |
| 47 | \( 1 + 5.09T + 47T^{2} \) |
| 53 | \( 1 - 7.61T + 53T^{2} \) |
| 59 | \( 1 + 6.70T + 59T^{2} \) |
| 61 | \( 1 + 7T + 61T^{2} \) |
| 67 | \( 1 - 9.38T + 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + 13.4T + 73T^{2} \) |
| 79 | \( 1 - 8.09T + 79T^{2} \) |
| 83 | \( 1 + 5.70T + 83T^{2} \) |
| 89 | \( 1 - 10.8T + 89T^{2} \) |
| 97 | \( 1 + 11.4T + 97T^{2} \) |
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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
−7.54022256928087440928849863933, −6.45230954958800040088864973951, −6.15759014921972241762739739026, −5.09522940414689261137001707323, −4.52953916624470647373469999962, −3.93267732805240572372565729720, −3.08801349628430757443220575255, −2.48609794171826852903278564832, −1.23307826625401211238811328081, 0,
1.23307826625401211238811328081, 2.48609794171826852903278564832, 3.08801349628430757443220575255, 3.93267732805240572372565729720, 4.52953916624470647373469999962, 5.09522940414689261137001707323, 6.15759014921972241762739739026, 6.45230954958800040088864973951, 7.54022256928087440928849863933