| L(s) = 1 | − 3i·2-s − 5·4-s − 7i·7-s + 3i·8-s − 9·9-s − 6·11-s − 21·14-s − 11·16-s + 27i·18-s + 18i·22-s − 18i·23-s + 35i·28-s + 54·29-s + 45i·32-s + 45·36-s − 38i·37-s + ⋯ |
| L(s) = 1 | − 1.5i·2-s − 1.25·4-s − i·7-s + 0.375i·8-s − 9-s − 0.545·11-s − 1.5·14-s − 0.687·16-s + 1.5i·18-s + 0.818i·22-s − 0.782i·23-s + 1.25i·28-s + 1.86·29-s + 1.40i·32-s + 1.25·36-s − 1.02i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.226904 + 0.961180i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.226904 + 0.961180i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 + 7iT \) |
| good | 2 | \( 1 + 3iT - 4T^{2} \) |
| 3 | \( 1 + 9T^{2} \) |
| 11 | \( 1 + 6T + 121T^{2} \) |
| 13 | \( 1 + 169T^{2} \) |
| 17 | \( 1 + 289T^{2} \) |
| 19 | \( 1 - 361T^{2} \) |
| 23 | \( 1 + 18iT - 529T^{2} \) |
| 29 | \( 1 - 54T + 841T^{2} \) |
| 31 | \( 1 - 961T^{2} \) |
| 37 | \( 1 + 38iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 1.68e3T^{2} \) |
| 43 | \( 1 + 58iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 2.20e3T^{2} \) |
| 53 | \( 1 - 6iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 3.48e3T^{2} \) |
| 61 | \( 1 - 3.72e3T^{2} \) |
| 67 | \( 1 + 118iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 114T + 5.04e3T^{2} \) |
| 73 | \( 1 + 5.32e3T^{2} \) |
| 79 | \( 1 - 94T + 6.24e3T^{2} \) |
| 83 | \( 1 + 6.88e3T^{2} \) |
| 89 | \( 1 - 7.92e3T^{2} \) |
| 97 | \( 1 + 9.40e3T^{2} \) |
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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
−11.84103708515514560740542888082, −10.78045684754044412180065163334, −10.39367448082955088525684755401, −9.173441904159672353729629000529, −8.054681153791945741295220390654, −6.59996025185774540769334403596, −4.91318249152499855118447356593, −3.64475528864649168192316555831, −2.45173404123421993236371897354, −0.58161847689952417523186842805,
2.76710226253761397332084242838, 4.90366327057276064386082294707, 5.74687065280699588611957894896, 6.63247088298899240676632274503, 8.026206915729551893618254016646, 8.550026610701262994721163564903, 9.649080825012350595319492991192, 11.18808143560634425999469611363, 12.12054891059381721710605056803, 13.40012745445837150239643834660
