| L(s) = 1 | − 9·3-s + 71.0·5-s − 29.0·7-s + 81·9-s − 634.·11-s − 676.·13-s − 639.·15-s + 1.87e3·17-s + 926.·19-s + 261.·21-s + 752.·23-s + 1.92e3·25-s − 729·27-s − 3.41e3·29-s − 5.68e3·31-s + 5.71e3·33-s − 2.06e3·35-s − 1.40e4·37-s + 6.08e3·39-s + 6.53e3·41-s − 1.44e4·43-s + 5.75e3·45-s − 1.83e4·47-s − 1.59e4·49-s − 1.68e4·51-s + 1.81e3·53-s − 4.51e4·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.27·5-s − 0.224·7-s + 0.333·9-s − 1.58·11-s − 1.11·13-s − 0.734·15-s + 1.57·17-s + 0.589·19-s + 0.129·21-s + 0.296·23-s + 0.616·25-s − 0.192·27-s − 0.753·29-s − 1.06·31-s + 0.912·33-s − 0.285·35-s − 1.68·37-s + 0.640·39-s + 0.607·41-s − 1.19·43-s + 0.423·45-s − 1.21·47-s − 0.949·49-s − 0.907·51-s + 0.0889·53-s − 2.01·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 9T \) |
| good | 5 | \( 1 - 71.0T + 3.12e3T^{2} \) |
| 7 | \( 1 + 29.0T + 1.68e4T^{2} \) |
| 11 | \( 1 + 634.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 676.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.87e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 926.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 752.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 3.41e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 5.68e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.40e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 6.53e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.44e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.83e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.81e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.67e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.74e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 6.18e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.01e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.02e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 4.61e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 7.44e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 6.72e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 2.92e4T + 8.58e9T^{2} \) |
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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
−11.00621183429790958441003864752, −9.984727139408641006634985471069, −9.651424692356673913148242680202, −7.953303368072560517680541495475, −6.93831702200783103194434698016, −5.45141903647962917126729190289, −5.27069836887509665911231197815, −3.08648563474828412910754993361, −1.75015828089621055832348474811, 0,
1.75015828089621055832348474811, 3.08648563474828412910754993361, 5.27069836887509665911231197815, 5.45141903647962917126729190289, 6.93831702200783103194434698016, 7.953303368072560517680541495475, 9.651424692356673913148242680202, 9.984727139408641006634985471069, 11.00621183429790958441003864752
