| L(s) = 1 | − 5·2-s − 6·3-s − 7·4-s + 30·6-s + 244·7-s + 195·8-s − 207·9-s + 794·11-s + 42·12-s + 169·13-s − 1.22e3·14-s − 751·16-s + 1.53e3·17-s + 1.03e3·18-s + 2.70e3·19-s − 1.46e3·21-s − 3.97e3·22-s + 702·23-s − 1.17e3·24-s − 845·26-s + 2.70e3·27-s − 1.70e3·28-s − 5.03e3·29-s − 3.63e3·31-s − 2.48e3·32-s − 4.76e3·33-s − 7.67e3·34-s + ⋯ |
| L(s) = 1 | − 0.883·2-s − 0.384·3-s − 0.218·4-s + 0.340·6-s + 1.88·7-s + 1.07·8-s − 0.851·9-s + 1.97·11-s + 0.0841·12-s + 0.277·13-s − 1.66·14-s − 0.733·16-s + 1.28·17-s + 0.752·18-s + 1.71·19-s − 0.724·21-s − 1.74·22-s + 0.276·23-s − 0.414·24-s − 0.245·26-s + 0.712·27-s − 0.411·28-s − 1.11·29-s − 0.679·31-s − 0.428·32-s − 0.761·33-s − 1.13·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.574274019\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.574274019\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 13 | \( 1 - p^{2} T \) |
| good | 2 | \( 1 + 5 T + p^{5} T^{2} \) |
| 3 | \( 1 + 2 p T + p^{5} T^{2} \) |
| 7 | \( 1 - 244 T + p^{5} T^{2} \) |
| 11 | \( 1 - 794 T + p^{5} T^{2} \) |
| 17 | \( 1 - 1534 T + p^{5} T^{2} \) |
| 19 | \( 1 - 2706 T + p^{5} T^{2} \) |
| 23 | \( 1 - 702 T + p^{5} T^{2} \) |
| 29 | \( 1 + 5038 T + p^{5} T^{2} \) |
| 31 | \( 1 + 3634 T + p^{5} T^{2} \) |
| 37 | \( 1 - 7058 T + p^{5} T^{2} \) |
| 41 | \( 1 + 294 T + p^{5} T^{2} \) |
| 43 | \( 1 + 7618 T + p^{5} T^{2} \) |
| 47 | \( 1 - 3020 T + p^{5} T^{2} \) |
| 53 | \( 1 + 626 T + p^{5} T^{2} \) |
| 59 | \( 1 + 30066 T + p^{5} T^{2} \) |
| 61 | \( 1 + 5806 T + p^{5} T^{2} \) |
| 67 | \( 1 - 12436 T + p^{5} T^{2} \) |
| 71 | \( 1 - 4734 T + p^{5} T^{2} \) |
| 73 | \( 1 - 14694 T + p^{5} T^{2} \) |
| 79 | \( 1 + 39804 T + p^{5} T^{2} \) |
| 83 | \( 1 - 41776 T + p^{5} T^{2} \) |
| 89 | \( 1 - 7970 T + p^{5} T^{2} \) |
| 97 | \( 1 - 78050 T + p^{5} T^{2} \) |
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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
−10.93268197945750802378450225353, −9.599092770832384613825554651340, −8.941140121557377958390470291700, −8.045372538600348489711077013736, −7.31329076298911122999566205902, −5.73780273324252552816554224473, −4.89483882811998413310341502422, −3.68206536575586481353975698950, −1.53357497037579532391402269917, −0.959457554031880286888127959311,
0.959457554031880286888127959311, 1.53357497037579532391402269917, 3.68206536575586481353975698950, 4.89483882811998413310341502422, 5.73780273324252552816554224473, 7.31329076298911122999566205902, 8.045372538600348489711077013736, 8.941140121557377958390470291700, 9.599092770832384613825554651340, 10.93268197945750802378450225353
