| L(s) = 1 | + 8.18·2-s + 9·3-s + 35.0·4-s + 73.6·6-s − 163.·7-s + 24.5·8-s + 81·9-s − 121·11-s + 315.·12-s + 728.·13-s − 1.33e3·14-s − 918.·16-s + 1.47e3·17-s + 663.·18-s − 341.·19-s − 1.47e3·21-s − 990.·22-s + 2.94e3·23-s + 221.·24-s + 5.96e3·26-s + 729·27-s − 5.72e3·28-s − 2.82e3·29-s + 5.16e3·31-s − 8.30e3·32-s − 1.08e3·33-s + 1.20e4·34-s + ⋯ |
| L(s) = 1 | + 1.44·2-s + 0.577·3-s + 1.09·4-s + 0.835·6-s − 1.26·7-s + 0.135·8-s + 0.333·9-s − 0.301·11-s + 0.631·12-s + 1.19·13-s − 1.82·14-s − 0.897·16-s + 1.23·17-s + 0.482·18-s − 0.217·19-s − 0.728·21-s − 0.436·22-s + 1.16·23-s + 0.0784·24-s + 1.73·26-s + 0.192·27-s − 1.38·28-s − 0.623·29-s + 0.964·31-s − 1.43·32-s − 0.174·33-s + 1.79·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(5.805762267\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.805762267\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 9T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + 121T \) |
| good | 2 | \( 1 - 8.18T + 32T^{2} \) |
| 7 | \( 1 + 163.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 728.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.47e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 341.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.94e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 2.82e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 5.16e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 4.25e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 5.87e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.66e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 5.98e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.75e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 8.24e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.34e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.94e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.46e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.30e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 8.72e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 9.05e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 8.62e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.30e4T + 8.58e9T^{2} \) |
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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
−9.462711133959990684106777519786, −8.687964286496715781721983266947, −7.55535220219727417119882670638, −6.54605773394832665828865868384, −5.94519538387149355925451774370, −4.97734210411734884338009577531, −3.77740572618489402182841570358, −3.33384124445459699353837456706, −2.46430470292177580876184550188, −0.855302903431712837654789115444,
0.855302903431712837654789115444, 2.46430470292177580876184550188, 3.33384124445459699353837456706, 3.77740572618489402182841570358, 4.97734210411734884338009577531, 5.94519538387149355925451774370, 6.54605773394832665828865868384, 7.55535220219727417119882670638, 8.687964286496715781721983266947, 9.462711133959990684106777519786
