| L(s) = 1 | − 3.25i·3-s + 73.9i·5-s + 112.·7-s + 232.·9-s + 575. i·11-s + 117. i·13-s + 240.·15-s − 223.·17-s − 1.75e3i·19-s − 366. i·21-s − 2.36e3·23-s − 2.34e3·25-s − 1.54e3i·27-s − 3.86e3i·29-s + 1.59e3·31-s + ⋯ |
| L(s) = 1 | − 0.208i·3-s + 1.32i·5-s + 0.869·7-s + 0.956·9-s + 1.43i·11-s + 0.193i·13-s + 0.276·15-s − 0.187·17-s − 1.11i·19-s − 0.181i·21-s − 0.930·23-s − 0.750·25-s − 0.408i·27-s − 0.853i·29-s + 0.297·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.666 - 0.745i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.666 - 0.745i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.51361 + 0.676744i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.51361 + 0.676744i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + 3.25iT - 243T^{2} \) |
| 5 | \( 1 - 73.9iT - 3.12e3T^{2} \) |
| 7 | \( 1 - 112.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 575. iT - 1.61e5T^{2} \) |
| 13 | \( 1 - 117. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 223.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.75e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 + 2.36e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 3.86e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 1.59e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 4.73e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 8.15e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 4.92e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 2.10e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.27e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 1.41e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 + 4.20e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 + 5.41e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 4.38e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.12e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 5.02e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 4.37e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 6.44e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 6.23e4T + 8.58e9T^{2} \) |
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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
−15.57228869940740915693637567202, −14.84535930821168705008994731390, −13.65670136627863458453981359873, −12.14803946239691533826786924394, −10.87926642551199050227714688080, −9.751953214143535884346290870259, −7.66147134868770394633779944625, −6.71610573358325930208941729867, −4.45266756426103196325240542233, −2.13819946288637623989555023354,
1.21401113985390081670342059595, 4.20531429105249763560298576626, 5.62557648523865038144034692479, 7.920009510989547500030322119696, 8.965111317482679452070749772155, 10.54519985194797584919174342405, 11.99240457682347783459635497562, 13.12246801330821239521998801843, 14.35133009338100467877881790601, 15.94691770462631846996029297219
