| L(s) = 1 | + 9·3-s − 20·5-s + 122·7-s + 81·9-s − 724·11-s − 914·13-s − 180·15-s + 1.00e3·17-s + 2.92e3·19-s + 1.09e3·21-s + 3.12e3·23-s − 2.72e3·25-s + 729·27-s + 6.74e3·29-s + 5.01e3·31-s − 6.51e3·33-s − 2.44e3·35-s − 5.27e3·37-s − 8.22e3·39-s + 5.23e3·41-s − 1.67e4·43-s − 1.62e3·45-s + 1.10e3·47-s − 1.92e3·49-s + 9.05e3·51-s + 2.20e4·53-s + 1.44e4·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.357·5-s + 0.941·7-s + 1/3·9-s − 1.80·11-s − 1.49·13-s − 0.206·15-s + 0.844·17-s + 1.85·19-s + 0.543·21-s + 1.23·23-s − 0.871·25-s + 0.192·27-s + 1.48·29-s + 0.936·31-s − 1.04·33-s − 0.336·35-s − 0.633·37-s − 0.866·39-s + 0.486·41-s − 1.38·43-s − 0.119·45-s + 0.0731·47-s − 0.114·49-s + 0.487·51-s + 1.07·53-s + 0.645·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.461248834\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.461248834\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p^{2} T \) |
| good | 5 | \( 1 + 4 p T + p^{5} T^{2} \) |
| 7 | \( 1 - 122 T + p^{5} T^{2} \) |
| 11 | \( 1 + 724 T + p^{5} T^{2} \) |
| 13 | \( 1 + 914 T + p^{5} T^{2} \) |
| 17 | \( 1 - 1006 T + p^{5} T^{2} \) |
| 19 | \( 1 - 2920 T + p^{5} T^{2} \) |
| 23 | \( 1 - 3124 T + p^{5} T^{2} \) |
| 29 | \( 1 - 6744 T + p^{5} T^{2} \) |
| 31 | \( 1 - 5010 T + p^{5} T^{2} \) |
| 37 | \( 1 + 5278 T + p^{5} T^{2} \) |
| 41 | \( 1 - 5238 T + p^{5} T^{2} \) |
| 43 | \( 1 + 16752 T + p^{5} T^{2} \) |
| 47 | \( 1 - 1108 T + p^{5} T^{2} \) |
| 53 | \( 1 - 22008 T + p^{5} T^{2} \) |
| 59 | \( 1 - 23716 T + p^{5} T^{2} \) |
| 61 | \( 1 - 45202 T + p^{5} T^{2} \) |
| 67 | \( 1 + 22756 T + p^{5} T^{2} \) |
| 71 | \( 1 - 53436 T + p^{5} T^{2} \) |
| 73 | \( 1 - 4790 T + p^{5} T^{2} \) |
| 79 | \( 1 + 1886 T + p^{5} T^{2} \) |
| 83 | \( 1 + 11268 T + p^{5} T^{2} \) |
| 89 | \( 1 - 73522 T + p^{5} T^{2} \) |
| 97 | \( 1 - 114154 T + p^{5} T^{2} \) |
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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
−10.28163791440371061148066800178, −9.802461667971217470035771083513, −8.402481028024005439472109006804, −7.76221408172325397144256472456, −7.19068824506501774344727618092, −5.23694016828121623549144872250, −4.90014977111138630826403684580, −3.22144437975741703168004117824, −2.36939151176486263101585443709, −0.817661817920907036156486434142,
0.817661817920907036156486434142, 2.36939151176486263101585443709, 3.22144437975741703168004117824, 4.90014977111138630826403684580, 5.23694016828121623549144872250, 7.19068824506501774344727618092, 7.76221408172325397144256472456, 8.402481028024005439472109006804, 9.802461667971217470035771083513, 10.28163791440371061148066800178
