L(s) = 1 | − 1.98·2-s + 3-s + 1.95·4-s + 2.85·5-s − 1.98·6-s + 7-s + 0.0979·8-s + 9-s − 5.67·10-s − 2.30·11-s + 1.95·12-s − 1.98·14-s + 2.85·15-s − 4.09·16-s + 2.52·17-s − 1.98·18-s + 4.60·19-s + 5.56·20-s + 21-s + 4.58·22-s − 4.10·23-s + 0.0979·24-s + 3.13·25-s + 27-s + 1.95·28-s − 2.68·29-s − 5.67·30-s + ⋯ |
L(s) = 1 | − 1.40·2-s + 0.577·3-s + 0.975·4-s + 1.27·5-s − 0.811·6-s + 0.377·7-s + 0.0346·8-s + 0.333·9-s − 1.79·10-s − 0.695·11-s + 0.563·12-s − 0.531·14-s + 0.736·15-s − 1.02·16-s + 0.612·17-s − 0.468·18-s + 1.05·19-s + 1.24·20-s + 0.218·21-s + 0.977·22-s − 0.855·23-s + 0.0199·24-s + 0.627·25-s + 0.192·27-s + 0.368·28-s − 0.497·29-s − 1.03·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.533900889\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.533900889\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + 1.98T + 2T^{2} \) |
| 5 | \( 1 - 2.85T + 5T^{2} \) |
| 11 | \( 1 + 2.30T + 11T^{2} \) |
| 17 | \( 1 - 2.52T + 17T^{2} \) |
| 19 | \( 1 - 4.60T + 19T^{2} \) |
| 23 | \( 1 + 4.10T + 23T^{2} \) |
| 29 | \( 1 + 2.68T + 29T^{2} \) |
| 31 | \( 1 + 0.917T + 31T^{2} \) |
| 37 | \( 1 - 5.93T + 37T^{2} \) |
| 41 | \( 1 - 4.15T + 41T^{2} \) |
| 43 | \( 1 + 10.6T + 43T^{2} \) |
| 47 | \( 1 - 0.601T + 47T^{2} \) |
| 53 | \( 1 - 1.41T + 53T^{2} \) |
| 59 | \( 1 - 4.87T + 59T^{2} \) |
| 61 | \( 1 - 13.4T + 61T^{2} \) |
| 67 | \( 1 - 14.2T + 67T^{2} \) |
| 71 | \( 1 - 12.3T + 71T^{2} \) |
| 73 | \( 1 + 10.8T + 73T^{2} \) |
| 79 | \( 1 - 6.67T + 79T^{2} \) |
| 83 | \( 1 - 17.4T + 83T^{2} \) |
| 89 | \( 1 + 10.8T + 89T^{2} \) |
| 97 | \( 1 - 5.15T + 97T^{2} \) |
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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
−8.567471855387023965222296076187, −7.992484226599239976451625252369, −7.44734093944585330708401326367, −6.57713553325663927244743610946, −5.61340443244297657396899421592, −4.96881070046994720776612292798, −3.69132296454768227698682789996, −2.47598791969015828740755424087, −1.90603055566908467158630259055, −0.910552318615700674476454273835,
0.910552318615700674476454273835, 1.90603055566908467158630259055, 2.47598791969015828740755424087, 3.69132296454768227698682789996, 4.96881070046994720776612292798, 5.61340443244297657396899421592, 6.57713553325663927244743610946, 7.44734093944585330708401326367, 7.992484226599239976451625252369, 8.567471855387023965222296076187