| L(s) = 1 | + 17i·3-s − 16·4-s + 49i·7-s − 208·9-s − 73·11-s − 272i·12-s − 23i·13-s + 256·16-s + 263i·17-s − 833·21-s − 2.15e3i·27-s − 784i·28-s + 1.15e3·29-s − 1.24e3i·33-s + 3.32e3·36-s + ⋯ |
| L(s) = 1 | + 1.88i·3-s − 4-s + 0.999i·7-s − 2.56·9-s − 0.603·11-s − 1.88i·12-s − 0.136i·13-s + 16-s + 0.910i·17-s − 1.88·21-s − 2.96i·27-s − 0.999i·28-s + 1.37·29-s − 1.13i·33-s + 2.56·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.4256445039\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4256445039\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 - 49iT \) |
| good | 2 | \( 1 + 16T^{2} \) |
| 3 | \( 1 - 17iT - 81T^{2} \) |
| 11 | \( 1 + 73T + 1.46e4T^{2} \) |
| 13 | \( 1 + 23iT - 2.85e4T^{2} \) |
| 17 | \( 1 - 263iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 1.30e5T^{2} \) |
| 23 | \( 1 + 2.79e5T^{2} \) |
| 29 | \( 1 - 1.15e3T + 7.07e5T^{2} \) |
| 31 | \( 1 - 9.23e5T^{2} \) |
| 37 | \( 1 + 1.87e6T^{2} \) |
| 41 | \( 1 - 2.82e6T^{2} \) |
| 43 | \( 1 + 3.41e6T^{2} \) |
| 47 | \( 1 + 3.45e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 7.89e6T^{2} \) |
| 59 | \( 1 - 1.21e7T^{2} \) |
| 61 | \( 1 - 1.38e7T^{2} \) |
| 67 | \( 1 + 2.01e7T^{2} \) |
| 71 | \( 1 + 1.00e4T + 2.54e7T^{2} \) |
| 73 | \( 1 - 9.50e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + 1.21e4T + 3.89e7T^{2} \) |
| 83 | \( 1 - 6.38e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 6.27e7T^{2} \) |
| 97 | \( 1 - 3.38e3iT - 8.85e7T^{2} \) |
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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
−12.74517407084250623819375037999, −11.61255330255000158007586559637, −10.39949357406696623372566339528, −9.882068079184456581102345115291, −8.783651952700924684966521810987, −8.357377339723924411560611051590, −5.84541391350072504502295340745, −5.08990311407056413090264769220, −4.12400070079137062066383297976, −2.93584000788115948774604698036,
0.18322272186938216413664157224, 1.20475748403600318982388483949, 2.95453367665585126965804070782, 4.75573542553034365660473373615, 6.11380649219034462579692243013, 7.27625325138586651737267955245, 7.933292905023312642315167536233, 8.956881689614059318660516929864, 10.32600946677662226251749735440, 11.58491129346635445740829955346
